Very similar to what I did the first time I visited, but I flew into Tokyo and out of Osaka. Spent 4 days in Tokyo and got the 7 day JR pass the day I left. Then did Kyoto for 3 nights, Hiroshima for 1, Miyajima the next day and on to Osaka for a few nights afterwards. Nara can be done in a day trip from Kyoto or Osaka. The three cities are all relatively close to one another. Try and get to Osaka at the weekend. There's a buzz to it on a Friday and Saturday.
Fuji is easy enough to get to. Probably want to go to Hakone with the Hakone Free Pass. There's a fair bit to do, but Fuji is notoriously elusive if there's any cloud so it's better planning for a day when you know you'll get the weather for it.