The unicorn's horn is about a metre long, as pale as ivory and as smooth as marble, with a spiralling twist from base to tip. Invested, in a credulous age, with mystical powers, it stands at a slight tilt in a Chamber of Wonders attached to Salzburg's main cathedral.

Of course, this narwhale's tusk never protruded from the head of a white woodland horse that could only be tamed by a virgin. But you'd keep that piece of myth-busting to yourself in the company of Wolf Dietrich von Raitenau, the most famous of Salzburg's iron-fisted church rulers.

Crowds flock to Salzburg in Austria for its Sound of Music and Mozart hoo-hah – and simply for its baroque splendour. But after a few days of sightseeing – Mozart's birthplace, tick; Do Re Mi fountain, tick – I'm keen to get to grips with deep Salzburg; with the forces and personalities that shaped it over 500 years of rule by a succession of powerful Catholic prince-archbishops.

And so after a sub-zero morning hike to the ramparts of the pale Hohensalzburg fortress atop the city's central Festungsberg mountain, I tramp back down, the path slippery with ice and the arches bearded with icicles, towards the city's old cultural heart. After the reward of a vanilla-filled bauernkrapfen, or farmer's doughnut, I make a beeline for the DomQuartier, slapping icing sugar from my coat as I go.

Open since 2014 to the public, this complex of princely private and state rooms, museums and galleries might not have the same allure as the Sound of Music, but it says more about Salzburg.

It's to Wolf Dietrich von Raitenau, ruler from 1587, that the city owes much of its style and its high cultural heritage. Von Raitenau levelled a swath of medieval buildings to make way for many of the palatial buildings of the DomQuartier, as well as five Italianate town squares.

Locals refer to Salzburg, cryptically, as the northern Rome, and the key to this riddle is von Raitenau's obsession with the baroque excesses he so admired in Rome during his time studying theology there. He returned and decided to remodel medieval Salzburg as a baroque showpiece.

Money was no obstacle. Salzburg enjoyed a lucrative monopoly in the mining and trade of salt; Salzburg literally means salt castle. Von Raitenau hired architects and artists from northern Italy – records put Andrea Palladio's successor Vincenzo Scamozzi in Salzburg in the winter of 1603-4 – and the Italianisation of the German-speaking city began in earnest.

Designed to inspire awe

I enter the DomQuartier's Residenz building through a triple-arched courtyard, its central niche framing a Hercules fountain, and find myself at once in the magnificent Carabinieri Hall. Built in 1600 for state functions and banquets, this 600-square metre space has rust red marble floors and white walls pierced by large windows. Awesome is such a commonplace that it's ceased to mean anything much, but there's no doubt that baroque-era visitors to the Carabinieri Hall would have awestruck.

Named after the mounted bodyguards introduced by von Raitenau in imitation of the Pope's Swiss Guard, the hall rises to a height of 12 metres. It's here that the prince's Italian obsession comes to vivid life, beneath a ceiling fresco of Neptune calming the elements to help Rome's legendary father, Aeneas, cross to Italy from Troy.

If it weren't for the snowflakes falling outside you could easily imagine you'd stepped from a sunny southern piazza into a radiant baroque palazzo. Salzburg's Catholic strongmen were as beguiled by power, prestige and luxury as any secular potentate (invoices reveal one bishop's 100 kilogram order for Milanese chocolate). Von Raitenau and his kind were priests who thought like kings.

My guide, art historian Maria Herz, stresses the "absolutist" nature of power wielded by the prince-archbishops who ruled Salzburg from the early 14th century.

"This complex of buildings had a double function," she explains as we stroll through the prince-archbishops' Residenz. "Here, at the centre of worldly power, they ruled as princes; in the cathedral joining the Residenz by a terrace they presented themselves to the city as spiritual leaders."

Von Raitenau lived a complicated life, Herz explains, with a mistress who went by the rather racy name of Salome Alt. "He presented Salome as his wife, more or less. She was very beautiful. She bore him 15 children, and he was loyal to her until the end. He asked the Pope several times for permission to marry her, if only to confirm her status legally, but it was denied."

The authorities in his beloved Rome doubtless feared the prospect of Salome and the family of 15 inheriting church property, and so refused to lend legitimacy to human reality. Plus ça change.

Alexander the Great

The prince-archbishops' worldly ambitions are teased out in a series of frescoes, paintings and stucco medallions decorating the staterooms and private apartments of the Residenz. They looked for a role model to Alexander the Great, ruler for a short while of the largest empire the world has known.

There are frescoes and paintings of Alexander cutting the Gordian Knot; Alexander taming his horse Bucephalus; Alexander in Athens; Alexander in Jerusalem. The ceiling painting in a lavishly decorated Knight's Hall, Herz explains, would have reminded visitors that Alexander needed to conquer other kingdoms because his own was too small.

As preparation for private audiences with the prince-archbishop – the ultimate goal for any envoy or petitioner – the visitor spent hours on his feet, passing through a sequence of progressively smaller rooms.

"The smaller the chamber the closer it was to the heart of power," says Herz as we walk from the lavishly decorated Audience Hall with its parquetry floor, gold-leaf stucco work and Belgian tapestries, and yet more Alexander myths, towards the private apartments. "The rooms are getting smaller, and the decoration more sumptuous."

The prince would often hold morning receptions from his bed.

As I pass beneath a ceiling fresco of Alexander half-asleep clutching a globe – the centrepiece of the prince-archbishop's bedroom – I pause to reflect on the ideal of absolutism passed from the young Macedonian king to Rome's Caesars, and then to Salzburg's prince-archbishops.

The fresco, by Salzburg's locally born and Italian trained Baroque painter Johann Michael Rottmayr, conveys the ideal prince who never sleeps and knows at all times precisely what is going down in his domain. It puts me in mind of Machiavelli, whom von Raitenau greatly admired. And perhaps, ultimately, Stalin.

The private apartments lead to a library reserved for secret audiences, its walls painted with maps of cities and regions viewed from mountain height and modelled, unsurprisingly, on those found at the Vatican museum.

A twist and turn through the DomQuartier brings me to the Salzburg Cathedral via an adjoining terrace supported by four arches linking the two main town squares. The route through the cathedral interior passes far above the floor, at Hunchback of Notre Dame height just metres beneath the gilded barrel vault. At this level sits the cathedral organist, the most famous of whom was undoubtedly the young Mozart.

Another turn gives onto an art gallery whose chief glory is a small Rembrandt of an elderly woman praying, followed by the Chamber of Wonders with its narwhale's tusk-Unicorn horn.

Chamber of Wonders

It's not the only wonder that was used by the worldly archbishops to wow guests with their dominion over nature: there are carved ibex horns, rare celestial globes, giant starfish, measuring instruments, a preserved blowfish, religious rosaries. In von Raitenau's Chamber of Wonders religion mingles seamlessly with science, or proto-science.

After two hours in the DomQuartier's interlocked spaces – halls, staterooms, bedrooms, a library, galleries, a cathedral, museum and treasury – I step out into a crowded market in Cathedral Square.

A violinist is packing up, fingers half-frozen. It's never too early on a freezing day for warm booze and the stalls selling glühwein are doing well. As I cross the bridge over the River Salzach I gaze back at Austria's wannabe Rome, its towers and domes and pitched roofs sharply etched against a pure white padding of fresh snow. Perhaps it's the onset of lunch that fires my imagination – I'm thinking of what power, ambition and imagination can achieve when seasoned with a pinch of salt.

Scene in Salzburg

• Kaffee-Alchemie, by the River Salzach, is run by a team of local coffee nerds committed to serving quality, fair-trade, single-origin coffee from recent-season harvests. The cafe's vintage feel, views of the river and hills, and well-chosen selection of cakes and brownies lend a nice vibe. kaffee-alchemie.at/?lang=en
• Cafe Tomaselli claims to be Austria's oldest coffee house, run by the Tomaselli family for more than 150 years. The interior, with its newspaper racks, crystal chandeliers, mirrors and inlaid nut-wood paneling, oozes tradition. So, too, does its famous pastry selection with more than 20 cakes, many of them polysyllabic tongue-twisters such as erdbeerschusserl (redcurrant jam, fresh strawberries, sponge, chocolate and whipped cream), schwarzwälderkirschtorte (black forest cake) and dobostorte (seven-layer Hungarian cake). A great location in the centre of town on the Alter Market brings crowds; for a quiet coffee choose the hour wisely. tomaselli.at
• Die Geheime Specerey is a centrally located, multitasking venue: a stylish delicatessen, a bar specialising in Italian and Austrian wines, and a restaurant serving great steaks, white tuna, artisanal specialties from its own meadow-raised pigs and fresh pasta. Better known among locals than tourists, it's the place to spot Austrian A-listers. felleis-knittelfelder.at/fkt/index_en.html
• Enoteca Settemila is a cosy wine bar serving platters of wine-friendly food – tapas, cheese and cold cuts – with a wine list emphasising the on-trend (lots of orange wine) and the local. enotecasettemila.at/
• Hangar-7 Take a break from the town's baroque glories and head to Hangar-7 at Salzburg airport. Run by Red Bull owner Dietrich Mateschitz, it houses the Flying Bulls aircraft fleet, a collection of F-1 racing cars, and a restaurant called Ikarus, which features a guest international chef each month, plus two bars, an outdoor lounge and a café. hangar-7.com/en/
• Imlauer Sky Restaurant, atop the six-floor Imlauer Hotel, has panoramic views of the old town and surrounding hills, indoor and outdoor seating, and a refined Austrian and international menu. imlauer.com/en/restaurants/imlauer-sky-bar-restaurant/

 Luke Slattery is the author of the historical novel Mrs M, published by Fourth Estate in November.