I believe that the Mt Aso volcano caldera on Kyushu (which I did as a day trip from Fukuoka, renting a car in Kumamoto) is still closed due to volcanic activity. You can possibly replace it with a day trip to Nagasaki, although I found the A-bomb memorials to be much more impressive in Hiroshima.

Fukuoka city itself has about one day's worth of stuff to see.