Closing the Arnold Schwarzenegger Museum leaves visitors with more time to spend exploring the hidden corners of the Altstadt, a UNESCO-listed labyrinth of medieval lanes, baroque façades and Renaissance courtyards constructed under the auspices of the Leopoldine Habsburgs before they decamped to Vienna in 1619. Located on the west bank of the River Mur at the foot of Schlossberg, the Altstadt is bisected by two main commercial arteries: Herrengasse and Sackstrasse. Among the inviting shops and restaurants, imposing historic edifices stand out, such as the baroque Palais Attems and the more visited regional parliament, the Landhaus. The latter houses the Zeughaus (Herrengasse 16, 801 79810, www.museum-joanneum.at), a huge historic armoury; it also has a spectacular courtyard, venue for concerts and, at Christmas, a nativity scene carved in ice. You’ll find more of these porticoed courtyards tucked away off Sporgasse, Hofgasse and Bürgergasse. These are the Altstadt’s finest medieval streets, to be found around the cathedral and the Mausoleum of Ferdinand II, the city’s last Emperor resident.

To view the Altstadt’s expanse of domes, spires and russet roofs, climb Schlossberg – either by funicular from Sackstrasse, by walking up the steps of medieval Sporgasse, or by lift from Schlossbergplatz. On the hill there are lovely walks among mixed woodland, scenic restaurants and Graz’s clock and bell towers – all that remains of the fortress destroyed in the Napoleonic siege of 1809. Looking down over the Mur, the bizarre ventricle-studded exterior of Kunsthaus Graz (Lendkai 1, 8017 9200, www.kunsthausgraz.at, closed Mon) comes into view. Resembling a human organ whose skin mutates at night with lighting effects, its visual effect inevitably eclipses the contemporary art it was built to house. And opposite, bobbing on the Mur’s turbulent waters, Brooklyn-based Vito Acconci’s artificial glass and steel island is similarly overwhelming.

• Tourist information: Herrengasse 16 (80 750, www.graztourismus.at).