Check out the 7 day JR pass with unlimited Shinkansen travel (not the fastest one, but the others are still pretty blooming fast :)). You need to book in advance from overseas.
I would visit Nara to feed the deer and see the giant Buddha. A previous poster mentioned a giant one near Tokyo too - that's Kamakura. That and Nikko (the see no evil, speak no evil, hear no evil monkeys) are good day trips.
In Tokyo, what kind of culture do you want to see? Do you want to get up super early to go to Tsukiji? Roppongi for the clubbing experience? Harajuku to see the mad fashions, Shibuya for shopping, shinjuku to get hopelessly lost in enormous crowds or Ameyoko market for the most insane market experience (not for the faint hearted)?
We did the Shinkansen from Tokyo to Osaka once, stopping in a seaside town called Atami, which looks like it was abandoned in the 50s and is full of hotels which will rent you a private spa bath (kashikiri onsen), nice if you're too shy or tattooed for the public experience.
In Kyoto, kinkakuji is beautiful, as is ginkakuji (kin and gin mean good and silver respectively). Don't miss Kiyomizudera, the beautiful temple set into the hillside.