The rain is very light in Kumamoto City right now but there are heavy rain warnings for other parts of Kumamoto, including the Aso area:
http://www.anshin.pref.kumamoto.jp/rireki/saigai/430000.html
This information came to me through an e-mail service which, unfortunately, is only available in Japanese.
I have some more information in English in the “Weather” section of my “Kumamoto Links” page:
http://www2.kumagaku.ac.jp/teacher/~masden/links.html#weather
The following page put out by the national government seems to have the best information in English:
http://www.jma.go.jp/en/warn/349_table.html
This information agrees with what I got in Japanese in that there are no warnings or advisories for Kumamoto City but heavy rain is predicted for most of the surrounding areas.
Unfortunately, I’m not aware of any e-mail warning system in English. As far as I can tell, you just have to go look at the page if you are worried about the situation. Please write a comment here if you know of a better way to get this kind of information.