From Fukuoka International Airport to JR Kokura Station
From Fukuoka International Airport, you can use (a) highway bus for Kokura Station, (b) subway and JR trains for Kokura Station, and (c) taxi and Shinkansen.
(a) By Highway Bus (Nishitetsu Highway Bus)
You can take Nishitetsu Highway Bus at Domestic Terminal 2 or 3. If you arrive at International Terminal, move to Domestic terminal 2 or 3 using free "Shuttle Bus" between international and domestic terminals.
(b) By Subway (Fukuoka City Subway) and JR Train
1. First, take Fukuoka City Subway from Fukuoka Airport to JR Hakata Station. The airport station of Fukuoka City Subway "Fukuoka-kuko" is in Domestic Terminal 2. If you arrive at International Terminal, move to Domestic Terminal 2 using free "Shuttle Bus" between international and domestic terminals.
2. Buy a ticket from "JR Hakata Station" to "JR Kokura Station" at "Train-Ticket Center" of JR Hakata Station. There are many trains from JR Hakata Station to JR Kokura Station. You can choose local train (about 100min), rapid service (about 70min) or express (about 45min). You can take the train of Kagoshima Honsen Line. Or buy a Shinkansen (bullet train) ticket from Hakata to Kokura (only 17 minutes). JR train fee are from 1,250 JPY to 2,200 JPY.
(c) By Taxi and Shinkansen (fastest but expensive)
1. Firstly, take a Taxi from Fukuoka airport (international terminal) to JR Hakata Station. Single fare is about 1,000-2,000 JPY. Running time is about 10min.
2. Buy a ticket of Shinkansen (Bullet train) from "JR Hakata Station" to "JR Kokura Station" at "Train-Ticket Center" of JR Hakata Station. You can take Shinkansen "Nozomi", "Hikari" and "Kodama". Single fare is around 2,200JPY (non-reserved). Running time is about 25min.