I don’t know how you’d like to sort out your itinerary, or what plans you already have, so mentioned here are just some ideas of interesting things to do around your dates. When I heard of a winter trip my mind went to a 5-day getaway from Tokyo into some more countryside areas, where the winter traditions are more alive than in the big city and very scenic with the snow.
The first day would be a day trip from Tokyo, without the need to change hotels, to see Mt.Fuji. In winter it is covered in snow and very beautiful.
The next 3-4 days you’d move your base to a city further north, where you’ll have easy access to the other sites on the itinerary. You mentioned wanting experiences over sightseeing, so I’ve come up with some things I think you might enjoy: meeting a geisha, visiting a sake brewery in Japan’s sake capital, staying at a hot springs covered in snow, a New Year’s festival and market, an overnight in a traditional Japanese inn and of course, lots of food! The market falls on the 6th and 7th of January, so ideally you’d be there during those dates, but if not I can see if they’re doing any other events nearby on different days.
You’d return to Tokyo on the fourth day, perhaps with an evening visit to see some winter illuminations.
Of course, this is just a first draft without knowing much about your group and interests, but if you’d be more interested in other things we could easily change them around.
Here are some photos from my own travels around the area, featuring some sites that would be included in the itinerary: