In reply to tingle:
We were there in March of last year and stayed in the technikerhaus while we waited for better weather.
We had a half day at Fulpmes, which was nice, the tree skiing saved the day as there was fairly thick cloud, another day we took the train to St Anton, which was big and fun. When the cloud lifted we headed up to Stubaier Gletscher, which was big and fun for the morning, before we went over the back to the Ambergerhutte.
The suggestion of staying a night in the dresdner hutte is an interesting one if you wanted a memorable night or two? The station at Innsbruck has different size lockers where kit can be left for a few days instead of keeping a hotel room.
I thought the stiftskeller was the best food, the imperial hofburg the best museum and the alpenzoo interesting.