Syracuse: Winter themed day trips?
Letchworth State Park
is called the "Grand Canyon of the East" and is well worth a visit, although I'm not sure about Christmastime.
Tokyo: Affordable hotels in Toyko
Have you heard of capsule hotels? They are popular with travelers who just need a place to crash for a few hours (and businessmen who stayed out drinking and missed the last train!) I think they run about 3000yen per night.