Do you have advisors within the study abroad and exchange department at your university? If you do, I would meet with them as soon as possible, it will save you a lot of work. There is going to be a lot of paperwork, and waiting, so it's good to have someone to communicate with regularly.
Although I do not know if scholarships would carry over or when you should start applying, I would see your exchange advisor (again, if you have one) and ask about the JASSO scholarship. Depending on funding, you can receive a pretty generous monthly stipend.