The point of entry in Japan is really up to you. On the contrary, if originating from Fukuoka, travel time would only take less than an hour by Shinkansen Bullet Train.
Mostly, tourists opt to stay in Hakata - noted as one of the contemporary areas in Japan or in other distant parts of Fukuoka and Kitakyushu. While others might travel from Tokyo, Osaka, Nagoya or other regions of Japan.
http://www.fuk-ab.co.jp/english/sp/cont.php?page=bus3
- After arriving from your international flight, hop on to one of its free shuttle service roving throughout the four terminals every 5-8 minutes.
Final stop would be at the Fukuoka Airport Terminal Subway Station.
Domestic Terminal:
- Whether arriving from Terminals 1, 2 or 3, upon reaching the first level, walk towards Terminal 2 and ride one of the 4 elevators to the basement floor leading to the airport subway station.
From Fukuoka Airport Terminal Subway Station, journey towards Hakata or K11, located along the Kuko Line for JPY 260 per person/way. Travel time takes exactly 5 minutes and just one station away from Fukuoka Airport Station. Once you disembark, transfer to one of Japan Rail transportation services bound for Kokura:
Option 1: Shinkansen / Bullet Train - the fastest among these railway cars going to Shin-Osaka (major stop) roughly at JPY 4,570 per person / way.
Option 2: Limited Express Sonic or Nichirin Trains bound for Kokura / Beppu / Oita / Miyazaki-Airport (major stops) for JPY 3,190 per person/way.
Option 3: Nishitetsu (Nishi-Nippon Railroad) Highway Express Bus from Fukuoka Airport to Kokura. Travel time takes about 1 hour and 30 mins. Bus fare costs JPY 1,230 per person/way.
Useful Links:
http://www.fuk-ab.co.jp/english/sp/cont.php?page=bus3
http://subway.city.fukuoka.lg.jp/eng/route/#b
http://english.jr-central.co.jp/index.html
http://www.nishitetsu.jp/bus/highwaybus/rosen/airkokura.html
http://www.fukuoka-airport.com/