Not on the Airport Line. Change at Hakata to the Nanakuma Line and get off at Ropponmatsu, Exit 3. The Dome Theater schedule decides whether the visit works.
Airport Line to Ohorikoen, about 15 minutes, then a 10-minute walk from Exit 3 or 6. Last admission is 17:00, so an afternoon landing decides whether it still fits.
The museum you want is Kyushu National Museum, and it is in Dazaifu, not central Fukuoka. Three routes: the Dazaifu Liner bus, Tenjin plus Nishitetsu, or a taxi.
Akasaka or Ohorikoen? The ruins spread across Maizuru Park, so the right station depends on whether you want the main keep area, the Korokan side, or Ohori Park.