Depending on your schedule I suggest to go for both:
1 day in Munich and some more time in the Austrian Alps.
For Munich I suggest to look for hotels in Feldkirchen, Dornach, or Riem. These are villages to the east of Munich inside the "Aeussere Ring" with easy public transport into Munich. Munich itself is always beautiful. You have many possibilities for sight seeing - the most famous the Rathaus at Marienplatz, Viktualienmarkt, Alter Peter and Liebfrauenkirche - all in easy walking distance. Additionally there are many great museums, like my favorite the Deutsche Museum (
http://www.deutsches-museum.de/ ).
For accommodation in the Austrian Alps you already have the right ideas. I suggest to contact the tourist office in Innsbruck (
http://www.innsbruck-tourism.at/ ). Depending on your finances I strongly suggest to consider camping as option besides hostels.