If you only have 2 nights, I would spend them either in Beppu or Kumamoto City. Depending on when you arrive in Fukuoka, you may not even have a full day. I often arrive in Fukuoka from Narita at 8 to 9 PM, which only gives me time to visit an Irish bar on Oyafuku-dori before going back to the hotel. If you arrive late, you would only have the next day to get somewhere and then to see something. If you stay in Beppu, you might take a day trip to Yufuin. If you stay in Kumamoto, you could take a day trip to Mt. Aso.From Hakata Station (Fukuoka) to Beppu takes about 2 1/2 hours by express.
From Hakata to Kumamoto City takes about 1 1/2 hours by train.
To get to Aso from Beppu it would be best by bus. This has been addressed at
http://www.japan-guide.com/forum/quereadisplay.html?0+65987
From any of the stations (Beppu, Aso, Kumamoto) you still have to get to what you want to see. If this includes going up to the crater on Aso, it will take a while or cost money for a taxi.
It takes about 3 hours by Shinkansen to get from Hakata (Fukuoka) to Osaka, so 6 hours round trip.
There are kankou bus tours of Mt. Aso from Kumamoto City, and probably elsewhere. You can find out more by checking out:
JNTO Mt. Aso acrobat file:
http://www.jnto.go.jp/eng/location/rtg/pdf/pg-704.pdf
JNTO Beppu acrobat file:
http://www.jnto.go.jp/eng/location/rtg/pdf/pg-705.pdf
