[Sangetsu]

You aren't going to see much in Hokkaido in December but snow and ice. And it's difficult to see all of the sights there unless you rent a car.

My favorite hotel in Hakone is the Fujiya. Their rates are a little above your price range, but they might be lower in the winter time. They also offer a discount to foreigners. The Fujiya is one of Japan's oldest hotels; Helen Keller, Charlie Chaplin, and John Lennon have all been guests.