Innsbruck, Austria: Historical town with student cool

Historical Innsbruck is quite the hoot. I'm amused by the way lovelorn ladies rub the bronze codpiece of a princely statue in the Maximilian Chapel. I like the clanging of church bells, the scowling presence of wobble-chinned Hapsburgs in palace portraits, the cafe-crammed square overlooked by its gold-leaf balcony, from which emperors gazed when Innsbruck was the late-medieval capital of Austria.

The town is encrusted with history, but what I like about Innsbruck too is its contemporary confidence. It has a youthful energy (university students make up a quarter of its population) and a liking for the new. It brings in star architects to create striking buildings. The angular white concrete of David Chipperfield's Kaufhaus​ department store is flanked by 16th-century apartments. The courtyards of Dominique Perrault's town hall in their skin of metal mesh and glass sit right in the gingerbread old town.

Zara Hadid​, who died earlier this year, designed Innsbruck's Bergisel​ ski jump, which sits like a hillside sculpture above town. Ski jumpers have fabulous tilting views of Innsbruck as they soar; visitors get a more sedate panorama from the cafe wedged at the jump's summit. This is my favourite monument in the venerable city, celebrating an improbable modern pastime. Locals like it, too. Later, they invited Hadid back to create stations for the Hungerburg funicular in her signature curved, futuristic style. Congress station sits right in town, an alien spacecraft that beams you up towards the 2269-metre summit of the Nordkette​, reached on a connecting cable car.

Italy is just over the mountains and Vienna far away. Innsbruck has a slight Italianate edge, not just in its pastel houses but in its love for sunny cafe terraces and fondness for shopping and sociable nightlife. Though mountains are all around, there's as much après as ski. The city has great bars, not least the circular glass aquarium of the 360° Bar, in which you can float like a goldfish above the light-twinkled city. It's one of several glamorous rooftop bars; another is 5th Floor atop The Penz hotel – part of the town hall complex – from which sloped old-town roofs and mountains merge.

There's a touch of the Mediterranean too in Innsbruck's restaurants, of which my favourite is the stylish Sitzwohl​, whose two chef-owners create classy versions of traditional Tyrol foods such as mushroom dumplings or spiced red cabbage, but also tempt with black gnocchi and turbot carpaccio. The restaurant is a cavernous space in a former school hall, yet has surprisingly intimate, semi-private dining spaces.

Sitzwohl might be the best dining in town, but I love Innsbruck's contrasts. You could do worse than linger over wine at Vinothek Dr Fischer, which couldn't be a more different space: a tiny bistro in which you can sling your coat over your chair back and chat with the eponymous owner. It too reminds me of Italy, with its bonhomie, red wines, antipasto platters and enthusiasm for forest-picked mushrooms.

You'll find the Vinothek in Wilten, a newly trendy district a walk down Leopoldstrasse from the city centre. Interesting boutiques have been popping up here of late, and hip eateries, of which cafe-bakery Deliris​ is a top recommendation for breakfast or brunch: creamy almond porridge, organic breads, olive-studded omelettes. It's only one of many emerging, creative eateries here, such as Robo​, inspired by world street food, or Immerland​, which takes the now tired cupcake concept to a new level.

Hovering between here and the town centre is Nala Hotel, which also subverts the Innsbruck stereotype of historical cuteness. Not one of its 50-plus rooms is entirely the same, but all share a witty decor in which antiques, trendy lighting and curious furniture collide. Some rooms push the limits; the all-white Snow White room isn't for those who shed black hairs. My room has a sink that has migrated from bathroom to hall, concrete shelving, and roll-back windows that open up to mountain views. The bathroom is aglitter in gold tiles, and I wallow like Marilyn Monroe, just happy to be here.