Hi Instruction01:
My wife and I live in Budapest and visit Vienna since its so close every now and then. But, each year in December we make our annual Vienna Christmas market trip to enjoy the awesome market in front of the beautiful city hall building.
Is it cold? Yes. In our years visiting the the temps (in celcius) have ranged from -1 to 6 for the highs and -5 to 3 for the lows. And it has both snowed and rained on us. But during the holiday season it gets a bit festive and no worries - there is always a coffee house somewhere in your vicinity. Staying 14 days will also allow you to explore areas outside of Vienna which could be really nice, especially if we're lucky and have some snow. (We've had a relatively warm winter this year with little snow).
One tip - be sure to stay in a well positioned hotel to wherever you believe you will be exploring the most. However, Vienna has an excellent transportation system, so you can always catch a ride to your destination.
Enjoy Vienna!
J. Foy