I think the most Japanese thing expats living in Japan can do for New Years is to go home. For the past 2 weeks, everyone's been asking me, "Aren't you going home for New Years? Don't you miss your family?" Over here, New Years is kind of like Thanksgiving, Christmas, and New Years, all rolled together. You're supposed to spend the few days before and after with your family. Furthermore, the whole country shuts down for the week: banks close, ATMs close, garbage pickup stops, supermarkets close, buses stop running, libraries close, gyms close. Pretty much anywhere you go on your own closes; only places you can go with your family, like movie theaters and certain restaurants, stay open.
Anyway, happy new year, congrats for making it here, and good luck for next year!
My New Year's resolution: use up and wash out jars of mustard and jam etc when I get close to finishing them, not leave them to molder in the fridge.