Japan Railways Shinkansen operates a train from Fukuoka to Hagi hourly. Tickets cost $28 - $40 and the journey takes 40 min. 2 other operators also service this route.
How to get from Fukuoka to Hagi
From Hakata train to Shin-Yamaguchi, train to Yamaguchi, then bus to Hagi. 4 h 1 min ¥6739. From Hakata train to Asa, train to Nagatoshi, then train to Hagi. 4 h 15 min ¥5175. From Fukuoka drive to Hagi. 2 h 9 min ¥2379.