Glad you're staying at the Goldener Adler. Make sure your kids order the hot chocolate!
Here's my top picks for the Innsbruck area:
1. Patscherkofel cable car. Take the city bus (or a taxi?) to the bottom. At the top: great hiking, friendly and photogenic cows, breath-taking view, light meals, beer and nature walk. Go only when the weather is clear; if it's extremely cloudy, it's not worth it.
2. Ski jump. The red hop-on/hop-on bus will take you there. Take the funicular to the top, enjoy beer, yummy pastries (excellent strudel) and the view. Even my three year-old was impressed with the death-defying slope of the actual ski jump (you can view safely behind glass) and wants to be a ski jumper when she grows up.
3. Folk art museum and Maximiliian's chapel. In the old town. Surprised by my toddler being completely impressed with the large statutes surrounding the mausoleum in the chapel. Considered one of, if not the best, example of 16th C statues. Plus your kids will get a big kick out of how Maximillian tried to claim more royal-ness by linking himself to Joan of Arc, King Arthur and St. George.
Not only did my daughter love these, but I remember 1 & 2 from when I was a kiddo and teenager visiting Innsbruck. I especially remember being disappointed as a kid when the weather was overcast.
In Vienna:
1. You should go to Prater Park for two things: Ferris Wheel (hope you have time to watch "The Third Man" in the next few days before you leave -- you laugh, of course) and the train ride. Buy a combi ticket for both. The other rides are skippable, typical amusement fare.
2. Schloss Schonbrunn. Try to get tickets to the Marionette Theatre. You and your kids can spend an entire day at the Schloss, like we did. The palace and grounds are spectacular, includes a maze.
3. Stephansdom, of course. I think you can climb the tower.
Feel free to email me.