Northern areas will be beautiful when it is covered with snow but transportation becomes difficult sometimes because of the weather. Skiing and snowboarding is very popular winter things to do for a Japanese since there are lots of mountains and ski resorts.
Also we like to soak into onsen on cold days. It warms the body so well and feels very nice and relaxing.
In Tokyo, some temporary skate links appear. Such as Tokyo Sky Tree and Akasaka Sacas. They let us enjoy skating in middle of the city (Just like NYC!)
Some foods are limited in winter: oden, Nikuman, hotpot dishes, and hot sake, etc.
These articles in Hub Japan may help you plan your trip :)
If you come to Japan in February,you can enjoy Sapporo Snow Festibal. For about a week. you can see several art of snow and ice at Odori park.