A few suggestions starting from Nagoya and Centair Airport.
Nagoya --> Inuyama --> Takayama (train) (If you like onsen Gero onsen is on this route.)
Meiji mura is an interesting side trip from Inuyama if you like Architecture.
From Takayama get a bus to Shirakawa-go (stay overnight in a traditional farmhouse) Bus to Kanazawa. From Kanazawa you can head north for off the beaten track.
It'll be cold in the mountains in December, and might even be snowed in.
Nagoya to Mie by Kintetsu. They have an excursion ticket available that is very good value.
http://www.kankomie.or.jp/sagasu11_traffic/t_06.html
Ise, Futami, Toba, Ise-Shima, Spain-mura (a bit desolate yet surreal in winter), You could also head to Nara and Osaka.
Koyasan would probably be spectacular in the snow.
Hiroshima has very pretty Christmas lights. After New Year there is the annual bugaku on nearby Miyajima.
Kagoshima and Sakurajima in Kyuushu are worth a trip also. (Milder weather in the southernmost part of the main Islands)
The other end of the country Sapporo has it's winter festivals at this time of year I think. I haven't been but friends have.
Again I haven't been but Sado Island could be interesting. But potentially severe weather on the West coast in December I would have thought.