I've heard cases that one may proceed to get a new passport and then show BOTH new passport & valid visa (on the expired passport) upon entry to Japan. Those two do not necessarily have to be on the same document.
In my country when we renew passport, the expired passport will be returned to us after a cut is being made on the cover (so that you will not be able to use it anymore), not sure if this is applicable in your case.
Anyway, I would still suggest you to verify if this is doable from the officials (e.g. the embassy) though.