There is more to see and do in Tokyo than I can describe in a single post. Really. I have lived there and visited more times than I remember, and I still haven't done everything.

Buy a city guidebook (several publishers have them) and a map, and figure out for yourself what your priorities are.

Finding reasonable food won't be a problem. Just pop into any one of the thousands (literally thousands) of family-owned restaurants in the city, most of which specialize in one category of food (noodles, tempura, donburi rice dishes, curry, sushi, etc.) If the prices aren't posted in the window or on a picture menu, you can't afford them.

As far as New Year is concerned, New Year's Eve is a family holiday in Japan, not a time for public merrymaking, so the main event will be going to the nearest Shinto shrine at midnight to pray for good luck in the coming year. The Buddhist temples ring their bells 108 times, once for each of the 108 sins that Buddhism recognizes. Ask the owners of your hostel for suggestions.

Some things are closed from January 1 to January 3. Check online before you go to a museum or store. Temples and shrines are all open, though, as people like to go and pray.