How long will you be in NRT?

If it's a short MR, I don't know what you can do. If you will be in Japan longer, you maybe able to get a notary public to prove citizenship while you await the passport. They will need a photocopy of your birth certificate. However, you may want to contact the airline and/or the Japanese embassy to confirm if this will be acceptable.

This works for travel to Mexico. As I mentioned, Japan and the airline maybe a different story, as the airline will be responsible for flying you back if you are not accepted into the country.