I received such a MEXT research student scholarship from October 2012 to March 2014. (In my case, I'd already made contact with a Japanese professor who I wanted to study with).
There's several things to consider in terms of how to contact people.
- You need to search Japanese university websites in your field. This is going to be nearly impossible if you don't have any Japanese fluency.
- Then you're going to want to find researchers who you could imagine working with presumably you'd narrow it in terms of both research interest and their ability to use languages you can use (e.g., English).
- Write them a letter explaining you are receiving a scholarship from the Monkasho (文部科学省) and would like to study in their university, starting from being a kenkyuusei (研究生).
- See who responds.
As a basic rule, you are more likely to get a response from professors at universities that are used to hosting international students with MEXT scholarships. This will mean national universities like: Tokyo University, Kyoto University, Tohoku University, Kyuusyuu University, Hokkaido University, Nagoya University, Hitotsubashi, etc. and well-known private universities like Waseda, ICU, Keio, Aoyama.
More minor schools (prefectural or city ones) will probably not be super familiar with the procedures and would require strong support and interest from the advisor -- who doesn't get much time or money to help work on this (and it's quite the administrative hassle).
Separately, Japan is a place where knowing people is more than half the battle so it's hard to build connections from scratch. Instead, you're hoping that at least one person will respond and either be open to hosting your or know someone who is that they are willing to forward your e-mail to.