I'd suggest to go into the city centre too - but stay away from tourist traps like the sacher (no offense, dia15222 - locals just thing differently)
For a typical viennese coffeeshop, try the "Landtmann" along the famous Ringstrasse. More info on current exhibitions etc. you can find at www.wien.info there are always lots of interesting things going on.

In case you prefer a nap, there is the NH Hotel Vienna Airport, right across the terminal - they usually offer very good day rates if you ask at the front desk.

If the weather is as freezing cold and snowy as it is today, I'd stay in the hotel - vienna is beatiful if the weather is nice, but really not too great on cold nasty winter days.