There are government grant schemes available. This is probably the most common one, on which I once wrote an advertisement (below). Contact me, anyway!
Dear colleagues,
Let me advertise JSPS Postdoctoral Fellowship, a postdoctoral fellowship for foreign researchers to work in Japan, funded by the Japanese government via the funding body JSPS. Webpage
Unlike common post-doc positions in which a principal investigator (PI) gets positions and calls for applications to fill them, an application for JSPS Postdoctoral Fellowship is made jointly by a foreign researcher and a host researcher in Japan. Your first step therefore would be to contact a prospective host researcher.
(So please note that I'm advertising not positions that I have, but a scheme that you can use with me or other researchers in Japan as your host. Note also that this is not an official advertisement by JSPS either :) I can only answer questions if I am to be your host.)
Some further notes:
In fact the common kind of post-doc positions (a PI gets them and he/she advertises them) are somewhat rare in Japan. Therefore many researchers, including myself, are waiting for your contact on an application for JSPS Postdoctoral Fellowship!
There are other funding opportunities for foreign post-docs as well; I myself would be very happy to seek opportunities for promising researchers with similar research interests.
Thank you, Ichiro