Yes, there is an imperial palace in Innsbruck as well, as the emperors of Austria liked to live there time and again with the absolute magnificent view from their chambers. It is also possible to visit these stately apartments (the giant Riesensaal is particularly breathtaking) afterwards stop off in the Café Sacher - which is also in Innsbruck too!
Located on the outskirts of Innsbruck, Schloss Ambras is easy to reach by sightseeing bus. Here the museum is in itself an exhibit with its original Renaissance chambers of art, because the castle is the oldest museum in the world. But to be honest: many only come because of the beautiful park and the local fowl. No one goes home without seeing the two white peacocks spreading their tails!
After the Hofburg Palace and Schloss Ambras in Innsbruck, several castles in the area are worth visiting. Just around the corner and close enough for a coffee in the main square is Hasegg Castle with its mint tower in nearby Hall. The "Münze Hall" exhibition (on the theme: "Money rules the world") is definitely worth a visit.
With around 132,000 inhabitants, Innsbruck is quite compact. However the great range of culinary delights gives both locals and visitors the feeling that they are in a metropolis.