For the case you'd like to follow your citizens (or alternatively to escape them for your New Year holidays), here's how travellers from different countries typically celebrate New Year and spend time on the holidays:
Russians: staying in Danubius Grand Hotel/Thermal Hotel Margitsziget, enjoing spa procedures for many hours a day
Hot wine/glintwein is sold everywhere in Budapest and every touristic destination, starting from 110ft in small towns to 250ft and more on the streets in Budapest.
Gellért Baths. NY06/07: on Dec 31, open until 1pm only.
Széchényi Baths. NY06/07: on Dec 31, open until 1pm only. On Jan 1, open from 8am.
Rule of thumb for dinner on Dec 31 is: either a restaurant is open all night (and is closed for NY program well before it starts), or it doesn't work at all on Dec 31 (and maybe even Jan 1). Therefore, you're very unlikely to find a dinner even for 7pm or 8pm without buying an overnight program.
Dinner prices range from 15.000ft (in restaurants at Liszt Ferenc tér) to 25.000ft (Rivalda on Castle Hill in Buda) and higher.
For Christmas and New Year days, there're food stalls set up at Vörösmarty tér offering skewer meat, pancakes, sauerkraut and grilled sausages, vegetable mix with good ratio of onion, and hot wine. In winter time, they "live" few minutes until get frozen, though.
In winter time, all touristic streets are full of grilled chestnuts.
For mass winter gatherings (like skating rink at Roosevelt ter), sweets are also cooked on the streets, like rolled kalacas (tubes of pastry rolled over a spill, and then grilled similar to doner kebabs).
Most restaurants/clubs at Liszt Ferenc tér have some program for New Year night (NY06/07).
Besides, the following restaurants have an overnight program (NY06/07):
Cafe Negro in Pest. NY06/07, all seats are booked before evening of Dec 30.
Rivalda in Buda. NY06/07: Set menu 26500ft includes welcome cocktail, 8 course dinner, champagne at midnight. Jazz trio all the evening.
Avocado Music Cafe & Restaurant in Pest. NY06/07: Starts about 8pm. In NY06/07, the party was coordinated by László Kállay, mobile 06-30-549-10-01, he was ultimate source of all party-related information for the guests.
Ground floor and gallery: Set menu 19900ft includes a welcome drink, 7 course dinner before midnight (each is small and light enough to not overfull you) plus 3 courses after 1:30am, if you stay so long. Visitors mix: mostly well over 30, majority are locals.
Downstairs: Buffet for 14900ft (13 dishes); place where DJ plays all the time, and most fun can be found. Visitors mix: bit younger crowd.
Before midnight: nothing happens outside people having their meals; surprisingly, Turkish pop music (not lounge easy-listening) is played all the time and is mostly annoying. Short period of belly dance.
After midnight: the dance starts at downstairs about 12:15am, disco music starting with ABBA and not going to anything more advanced.
The following restaurants are closed: Malomto in Buda-Dec 31..Jan 1 (NY06/07).
Buda Castle Labyrinth Minimalheadz New Years Apocalypse . NY06/07: Entrance 2500ft in advance, 3500ft upon arrival. Several dance halls right inside the labyrinths, each hall with its own DJs. Starts at 21.
Gárdonyi Géza Theater. Jan 1, 7pm (NY06/07): New Year Concert--popular classics; digestible modern dance; some laser show. The conductor jokes and plays with the audience (even makes it sing with the soloist), but all in Hungarian--so better have a friend to help you. Master of ceremonies resembles provincial soviet-time song contest. Performance is introduced by mayor of the town.
The concert attracts the creme of the town--therefore evening dress is strongly recommended. At the entrance, welcome champagne is served.
Pátria Panzió: Book several apartments if you travel with a company; ask if you can use the pension's private patio exclusively for your gatherings and time together--alternatively, reserve a room in one of the apartments for your company gatherings in the evening.