Hi Grafster-
I'm living in Japan now, so happy to offer some suggestions.
A week and a half, eh?
Depends what you're into but I recommend spending most of the time around Kyoto.
Tokyo is good for shopping and wandering, but doesn't need more than a few days - must's are Asakusa Senso-ji/Meiji-jingu in Harajuku if you like shrines/temples; Harajuku, Shibuya, and Akihabara for shopping/good people-watching, going up a tall building for the view (I reccomend Mori tower in Roppongi Hills, also an art museum). If you're into music check this out for live stuff:
http://www.tokyogigguide.com/ Riding the yamanote line during rush-hour can also be an interesting experience. If you like sushi there's some great sushi bars around the Tsukiji fish market area.
I hear Kamakura is also a nice day trip from Tokyo, plenty of tourist sites there, or if you are coming in the summer you can try your luck hiking Mount Fuji, also near Tokyo.
Kyoto has lots more in terms of tourist sites - plenty of beautiful temples and shrines, etc. and the city is a much more manageable scale than Tokyo. And you definitely need at least a few days there to see it all.
Osaka is fun also, great food. And spa world
http://www.spaworld.co.jp/english/service.html
is epic if you're into onsens.
Nara is great, too (can be a day trip from Kyoto or Osaka) - the beautiful old Todai-ji temple and the semi-tame deer that walk around the park!
Further afield and worth a bit of time is Hiroshima and Miyajima which has a beautiful shrine on an island.
I'm in the north of Japan but can't say there's anything too amazing up here, unless you're coming for skiing or hiking but even then the more central Nagano region, or (farther away) Hokkaido might be more interesting.
Depending on when you're here, I also recommend trying to see a sumo event, a baseball game, or local festivals - check out
http://www.japan-guide.com/ for more info
Things can get expensive here but you can keep it cheap with some luck. Convenience stores (called conbini) sell an array of cheap foods that are great for meals on the run. Depending how much you want to travel around, consider getting a rail pass
http://www.japanrailpass.net/eng/en003.html which can definitely be worth it if you want to get places fast via the shinkansen (bullet train).
For cheap accomodation check out:
http://www.couchsurfing.org/
or
http://www.hostelworld.com/
Good luck and have fun!