Vienna is a beautiful city. I visited in January last year and though it was cold it was dry so just be sure to wrap up well.
Places to visit include Schombrum(sp?) palace, Schloss Belvedere, the Spanish Riding School (when I was there the horses were on holiday so you just got to visit the show theatre and stables but didn't get to see a performance. Not sure if this happens in Jan every year).
€3 tickets can be bought for the opera if you're prepared to queue. Even if you don't go to a performance you can do a tour of the opera house which is very interesting.
You can take a tram around the Ringstrasse and see all the fabulous buildings. They have a skating rink in winter which could be fun.
Don't forget to visit the Sacher hotel restaurant and have some of their famous Sachertorte. At least one visit to a coffeehaus is a must, cakes are delicious especially the apple strudel.
There are lots of concert halls with performances of Strauss & Mozart. I went to one and the standard was amazing.
I stayed in
http://www.lindner.de/de/LHAB and found it very clean, friendly,reasonable and very convenient for transport.
The Grand Hotel looks amazing and is very central. If you want luxury it looks the business.
You'll have a wonderful time. Enjoy.