The Castle can be visited
Innsbruck Castle

Innsbruck, Tyrol, Austria, is home to the Hofburg, a palace which once served as a residence for the regional princes. Its current appearance, in the style of Viennese courtly Rococo, is an architectural point in the city's history.
History+
The origins of the Hofburg date back to the Middle Ages. Duke Leopold III used an initial lodging on the site, whose properties were expanded by his son Friedrich IV. Archduke Sigismund the Rich had a medieval castle built here, which was later expanded under Emperor Maximilian I. A fire in 1534 destroyed the ceiling of the Giant's Hall, which King Ferdinand I had restored. Earthquake damage in 1536 led to the reinforcement of the facade with semicircular towers.
Source: Wikipedia · Licence: CC BY-SA 4.0 · Image: Robby-BF
This description draws on the facts documented in the linked Wikipedia article; Wikipedia content is licensed under CC BY-SA 4.0. Wording and editing: Mobile History.
Location: 47.2689, 11.3944