The problem is; your host institution has to jump through a lot of hoops to bring you legally to Japan. There is no "intern" visa, and no company that I know if would want to invest the time and effort to hire a foreigner just for a month or so.
Is the working holiday visa an option for you? If so, you could come to Japan on that and try to find casual work.
I typed into Google the following keywords:
internship, Japan, October 2016
I got a bunch, but in looking at the first half a dozen, the deadline dates for applications are gone. Pretty much what I already told you. (One had a Sept. 4 deadline, so you JUST missed it.)