Okay, then what?
First if you are arriving by plane, go to the information office in the arrivals hall and get a 24 hour TravelCard; if you dont, then the hotel will sell you a 24 hour BudapestCard for twice the cost. Maybe the hotel will sell you individual tickets but i doubt it. There are vending machines in all the metro stations and at some of the tram stops (cant tell you which). But you will need one or the other to make getting around easier.
It gets dark fairly early in Budapest in December. You can look up the actual time; but its around 5 or 5:30. If your flight arrives at noon, you can be on the tram by maybe 1:30 or 2:00.
The food at the market is fairly authentic, but not the greatest; but it can be a fun event. Maybe you just snack a bit and wait for dinner?
Once done at the market go back out front and you will see a tram parked there. That the 47/49 Tram. If you get on it moving away from the river, sit on the right side facing backward you will see the great synagogue just after the second stop. The next stop is the end of the line. Get off and walk across the street (to the same side as was the synagogue), turn left up the sidewalk and in about 2 blocks you will be at the end of Andrassy ut. (the Av. des Champs-Élysées of Budapest).
I am going to suggest you walk up Andrassy ut until you are tired of walking. There are restaurants, great shops, fantastic old architecture and the Opera House. I want you to be here when it gets dark, cause its beautiful. As the sun sets, hop on the M1 metro line. There are stops all along the way as it runs directly under the street. You want an entrance that says Hősök tere so you know you are getting on it in the right direction (entrances on the right side of the street as you walk up). Get off at Hősök tere (Hero's Square). This is also worth seeing at sunset an later.
If you have the energy, walk back down Andrassy ut a bit to see some of what you missed on the way up. This end is mostly mansions and embassies.
Try and pace it so that at 6:30 to 7:00 you are at the end of Andrassy ut. That metro stop is called Bajcsy–Zsilinszky út. At the end turn up the sidewalk to your right and in about a block and a half you will find Belvárosi Lugas Étterem http://lugas-etterem.com/etlap/ Did i mention to make a reservation? Do so. Good solid Hungarian food in a casual atmosphere. Good service. Best of all easy to find and on the route.
After dinner you have two choices. Walk to your hotel or take the M1 to very near your hotel. If you were to draw a line down the center of Andrassy ut all the way to your river, it would almost hit your hotel. Walk the back streets back and forth till you are there. Great walk, lots of shops and architecture. Along the way you will pass Vorosmarty ter which is ground zero for river boat tourism AND the Vorosmarty ter Christmas Market will be open.
If you take the M1 you will enter the stairs that say Vorosmarty ter and stay on to the second stop. Up the stairs and you are in Vorosmarty ter Walk about a block to the river and about a block to your right and you are home. The river front is beautiful in evening.
The weather in December is Very, Very Cool to downright Cold. But its a life style in Budapest and it doesnt stop much. Some how its more comfortable when you see the number of people around you seemingly unaffected by it.