If you do hit Japan in that "can only go once" manner, you'll obviously want to hit Tokyo and Kyoto, but if you're interested in Japanese culture as a whole, I would very seriously add Hiroshima to those as an absolute must see, even if only for a day or two. I've never gotten around to posting any of the posts I've written about Hiroshima (none of them seem to get it quite right), but I think it gives an essential view on contemporary Japanese culture.

Depending on how long you're there for, getting a look at the countryside et al would also be good. If you're doing a round-the-world bit, you could jump from Hiroshima (or more likely Fukuoka which is pretty near by) over to Korea pretty easily. So --> Hawai'i --> Toukyou --> Kyouto --> Hirosima --> Korea sounds like a good trip.