Japan Society for the Promotion of Science (JSPS) provides postdoc positions for foreign (non-Japanese) researchers. You can stay in Japanese institutes for 1-2 years using this program. See this page for the detailed eligibility.
There are two opportunities for the application in a year. If you want to start your stay between September and November, the deadline of the application at NAOJ is usually in early April. If you want to start it between April and September, the deadline is usually in early August of the previous year. See this page for the details. Please note that the application deadlines at NAOJ are different from those shown in the JSPS web page.
You have to select a host researcher to apply for this program. The professors in the list can be the host. You can select Full, Associate or Assistant professors. The application needs to be submitted by the host researcher. Please contact a possible host researcher when you wish to apply for this program.