1 day in Innsbruck

Neuschwanstein castle may be relatively near as the bird fly, but it takes a long time driving and by public transportation probably one full day, let alone an afternoon, would not be enough.

Innsbruck has a recently restored Hofburg (Court palace) with a main hall mimicking those at Schoenbrunn. And a (German) Renaissance castle on the hills (Schloss Ambras) with a large park and a couple of small museums. You barely have time for both.

We went to Neuschwanstein by rental car one day as a day trip from Innsbruck. Not worth the effort to me.
Innsbruck has a rather compact city center with some small museums, etc. What's the draw is the incredible views of The Alps. The city is in a valley about a mile wide--with two sets of 10,000' mountains above your head. You can see maybe 50 mountain peaks from the city. They have ski lifts 12 months of the year that take you up to Haflekar where the views are especially incredible.
We love getting up on the mountain sides rambling from village to village (by rental car) looking at the mountain vistas and agricultural sights. If you stop in one of the village restaurant pubs, you'll feel the people to be especially warm and friendly. Buy the bar a round of beer, and it's the last money you'll ever spend all night.

Go up by the "Hungerburg" train and further up by the cable car and enjoy the view from above.

The more and more I think about it however much I want to see the castle doesn't look like its easily possible from Innsbruck.
I love mountains and country side . Think I will just go with the option that wmt1 suggested.
If anybody has any other ideas . Please tell.