My husband and I stopped in Innsbruck a day in early February, we were traveling from Munich to Castelrotto. The train station was busy, but clean and organized. There are a lot of luggage lockers, a nice woman helped us with the instructions. Basically you closed the door, inserted money (3.50 Euros, it gave change too) and a large plastic token was dispensed. When you return, you deposit the token and your locker opens.
We walked down the Maria-Theresien-Straße to the Golden Roof. It was a Sunday morning and nothing much was open before 10:00, we did find a restaurant that was setting up so had a coffee and little snack. We did go up in the Stadtturm and had lovely views all around. It was a nice morning so we walked to the Bergisel ski jump. I had made email reservation for lunch to the Sky Restaurant, tell the person at the ticket desk and you will get a discount on admission to the ski jump. We really enjoyed this, give yourself 30-45 minutes to look around and then take the funicular up to the viewing deck and restaurant. We walked back to the train station, about 30 minutes.
It's a great town to spend a few hours.