I am not a resident but I have been to Korea during new years and I think most of the bars and restaurants opened during new years. Convienent stores were definitely open so you won't have trouble finding emergency stuff like asprin, tissues or whatever.

recomendation:
I heard the spa in W hotel Seoul is absolutely amazing. Its suppose to be really beautiful and luxurious. They offer spa for men too. But good things dont come cheap; its a bit pricey so be aware.