The Christmas markets are exceptional and they run from mid November until about 2pm on Christmas Eve. Then open again the next day, but without the food stalls. So, if you can, arrive a few days before Christmas Eve or you will miss something special.
I brag on Budapest a lot, but to be fair, Budapest doesn't throw the greatest New Year party. For that I would run up to Vienna. Given a choice I would arrive earlier in Budapest for the Christmas Markets even if it meant missing new year.
Szechenyi Baths, Yes, its open 365 days a year, but with shorter hours I believe on Christmas Eve/Day and New Year Day. Details here: http://szechenyispabaths.com/szechenyi-bath-christmas/
Chain Bridge. Its a public road. Yes it will be open
Funicular. Pretty sure its open. More detail here https://www.bkv.hu/en/buda_castle_funicular_/open_funicular_en
Buda Castle. Well the district will be open, the museums? Hours are here http://budacastlebudapest.com/christmas-buda-castle-budapest/
Fisherman's Bastion. Again, a public area. The access to the upper deck may be closed or it might be free. Maybe that previous link has some information.
St. Istvan’s Basilica. Midnight mass on Christmas eve, probably services on Christmas day. Otherwise you might find something here http://en.bazilika.biz/
Parliament. I would bet it will be closed 24, 25, 26 and the first. Maybe you can find something here http://latogatokozpont.parlament.hu/en/opening-hours