While there is no long-haul flight, Fukuoka International Airport is directly accessible from Korea (Seoul, Busan, Jeju), China (Shanghai, Beijing, Dalian, Qingdao, Guangzhou, Shenyang), Hong Kong, Taipei, Manila, Bangkok, Ho Chi Minh, Singapore, and Guam. Note that not all destinations are served on a daily basis.
From Fukuoka International Airport, JR Shinkansen Hakata Station is accessible in just 5 min - a 250 JPY subway ride. But you need to take a free shuttle bus (runs in every 8 min.) from the international terminal to the domestic terminal where the subway station resides. So it may be convenient to take a taxi to get to the Shinkansen Hakata Station (10 min. / about 1,300 JPY) directly from the international terminal especially if you have a big baggage.
From the Shinkansen Hakata Station, you can take "Nozomi" or "Hikari" train to JR Hiroshima Station (70 min. 8,900 JPY based in reseaved regular seat, one-way; additional 3,490 JPY for a green-car seat). Departure time from Hakata Station is as follows:
06:00, 06:04+, 06:30*, 07:00, 07:04+, 07:30*, 07:50, 08:00, 08:04+, 08:30*, 08:37+, 09:00, 09:04+, 09:30*, 09:37+, 10:00, 10:04+, 10:30*, 10:37+, 11:00, 11:04+, 11:30*, 11:37+, 12:00, 12:33*, 12:37+, 13:00*, 13:30*, 13:35+, 14:00, 14:04+, 14:30*, 15:00*, 15:04, 15:30*, 15:50, 16:00, 16:04+, 16:30*, 16:35, 17:00, 17:04+, 17:30*, 17:41+, 18:00, 18:04, 18:25*, 18:54*, 19:00+, 19:30*, 19:35+, 20:00*, 20:04+, 20:19+, 20:40, 21:11, 21:46=Note: * operated by N700-series car
Note: + Hikari (8 cars, no green car)
Note: = Kodama (stops every station, omitted except last train)