I can't advise on lodging, as I've got friends, but for a day, go to the city center and wander around the Marienplatz area -- pedestrian zone, Frauenkirche, St. Peter's, Viktualien Markt, etc. Go to a cafe and have some coffee, wander around more...

If the weather isn't cooperative (or even if it is), there are a number of good museums. If you are into technology, the Deutsches Museum is first-rate. For art buffs, the Pinakotheken - Old, New, and Modern are fabulous. I have a soft spot for the Lenbachhaus, which has a great collection of turn of the century (19-20) and 2oth Century art. If you are into history, close to the Marienplatz is the Residenz -- the old royal residence, which is quite nice and also has the Schatzkammer or treasury with lots of valuable stuff that can be nice to look at. Nyphenburg Palace is another royal residence that isn't as central, but has a lovely park surrounding it and is still easily accessible from the city center -- the public transit system is great.

For the evening, maybe go to a beer hall and make some new friends or perhaps to go a concert if you like cultural stuff -- there is never any shortage of highbrow and lowbrow entertainment.

Enjoy -- Munich is one of my favorite cities in the world.