The question was in the Austria Forum https://community.ricksteves.com/travel-forum/austria/new-year-s-eve-in-vienna but I thought the answer made more sense here.
We are spending Christmas and New Year in Budapest again this year so I did some research that I thought might be helpful if you are thinking of going to Central Europe
Since your request for hotel information followed my suggestion that you might look towards Budapest as well let me suggest the K&K Opera Hotel witch fits your budget. http://www.kkhotels.com/en/hotels/budapest/k-k-hotel-opera/welcome/ nice 19th century building next door to the Opera House and Andrassy ut.
Here is an idea where you don’t have to worry about rowdy crowds and broken glass (did New Year’s in Rome a few years back and experienced the same so it’s not unique to Vienna by any means) http://ppcdn.500px.org/4932259/468b7473e2387eb183fd6b5c5c4cf53a91f968d2/5.jpg
Ice Skating at Városligeti Műjégpálya. http://budapest-card.com/files/partners/109.jpg
Bath Houses outside (note the snow) http://guidepal.blob.core.windows.net/inspireimage/inspire103057.jpg or inside a 14th century bath house http://3.bp.blogspot.com/__hpVNPe3HNw/TMMu88ToBfI/AAAAAAAABKI/qiWLg2I5AA4/s1600/WEB+rudas+0.jpg or maybe you prefer classical: http://www.thevandallist.com/wp-content/uploads/2013/05/Gellert-Baths-@-Budapest.jpeg with the dubious history of having been visited by Himmler and Schindler. Even bathhouse party night http://attractions.topbudapest.org/wp-content/uploads/2011/06/Design-Szechenyi-Bath-Sparty-Budapest-Spa-July-27-2013.jpg The Bath house New Year Eve part is strangely on the 30th of December http://szechenyispabaths.com/szechenyi-baths-new-year-bath-party/
Concerts New Year’s Eve, Before and After
Opera New Year’s Eve Gala Performance; http://www.worldwideticketing.com/images/opera-ballet/hungarian-state-opera/hall.jpg and http://www.opera.hu/en
“Tourist” Concerts http://budapestchristmas.com/budapest-new-years-eve-classical-music-concerts
Liszt Academy (fantastic venue) http://zeneakademia.hu/en/home and http://zeneakademia.hu/en/lisztery/video/-/asset_publisher/fCa86eGLCFdM/content/a-2013-oktober-22-i-unnepelyes-megnyito-gala
They run through 31 December in Budapest http://cdn-wac.emirates247.com/polopoly_fs/1.434344.1324785528!/image/744982653.jpg
Shopping in the three story 19th century market hall http://i11.photobucket.com/albums/a171/danperry/Hungary/Budapest/The%20market/1-IMG_3889.jpg
And the holiday lights http://2.bp.blogspot.com/--tIpbcA5HcM/TvDqWvaYXsI/AAAAAAAAAcg/n1SsL9mejgE/s1600/1828771-Christmas_lights_on_Andrassy_Avenue_Budapest.jpg and even the trams http://blog.utazol.hu/wp-content/uploads/2013/11/Budapest-KAr%C3%A1csonyi-vasar-2013-Zsolt-Andrasi.png and http://media-cache-ak0.pinimg.com/736x/f6/0f/ff/f60ffff302046f46a437f676f64b2f4b.jpg
Here is the long list: http://www.budapestbylocals.com/new-years-in-budapest.html#free