In Tokyo you can see asakusa, which is a beautiful complex of temples and shrines. Also kamakura, a day trip from Tokyo is great for temples and they have a giant Buddha. I stayed in Tokyo around 5 days with 1 day in the countryside kayaking and visiting a Sake factory. And it was enough time to see a lot. Can't give you any advice on the rest if japan unfortunately. But a friend tells me bullet train tickets between Tokyo and Osaka can be around £150 so try and book in advance or check for cheap internal flights if you are short of time.
Also the rate of the yen is low. So now is the best/ cheapest time to visit we ate out in nice restaurants for under £10 and a sushi place with 4 dishes and a beer was around £6.