Don't worry about it. Just put any Japanese address in there. I've taken a lot of flights out of here, and the airline staff could care less where you live, only look at your passport and usually make sure you have entry permission into your destination country (I fly a lot to China, so visa required).

Here, use this:
〒108-8282
2-16-1 Konan, Minato-Ku
Tokyo (Enter that in the "State" dropdown)

Phone 03-5783-1111

That's the address for the Intercontinental Strings Hotel in Tokyo.