Got back from Japan, revising what I said of Osaka. Still less to do than Tokyo and less historical than Kyoto but its fun for a day or two especially if Kyoto is thunderstorming or dead. Its only like 45 minutes away from Kyoto station, so I'd still just recommend staying in Kyoto and taking the train in.
Idk if you've made housing arrangements yet, but my friend and I used Airbnb the whole way and couldn't have been happier. Wound up being about $40/night each in both Kyoto and Tokyo.
If you want to treat yourself, the best meal we had was at Ginza Maru (
http://www.maru-mayfont.jp/ginza/) in Tokyo. They didn't speak a lot of english but it didn't rly matter cos the prix fixe was the best meal I've ever had. Don't just take my word for it; the Japanese food critic for the Times recc'd it to us.
To shop in Kyoto you should really just zigzag through the Nishiki market and the streets around it from like 5-10PM. I can't remember any of the stores I went to (except Kapital, which was cool) but there were so many unnamed vintage and small brand boutiques that I just had to stop checking them all out. Food is definitely the highlight though; there's near infinite stores selling kababs of anything you can imagine, beer and junk food. There's also this great lil ex-pat bar we found called the Fishbowl run by this cat Jesse who's funnier than just about anyone i've ever met. Go there if you miss hearing ppl speak english. It's east of Nishiki with all the other bars/nightlight.
Go to the Kyomizudera temple and go down into the Buddha womb its surreal.
Also idk what arrangements you've made yet but you should spend at least a couple days in the countryside. We stayed a few days at a bathhouse in the mountains with meals included and long hiking trails and like 10 onsen and it was the most relaxed i've ever been.
Also climbing mt. fuji is awesome but intense. you should do it if you have time but wear boots and bring water because its for sure a mountain.