Fukuoka Airport is located to the east of the city, surprisingly close to the city centre (the domestic terminal is only 2 subway stops away from the Hakata JR station - there is a ~10 min. free bus connecting the intl. terminal to domestic). Within the country, Japan Airlines and ANA fly to Fukuoka from most larger cities, including Tokyo (both Haneda and Narita), Osaka (Itami and Kansai), and Nagoya (Komaki and Centrair Airport). There are scheduled flights to most major cities in China and South Korea, as well as Taipei, Hong Kong, Singapore, Bangkok, Manila, Hanoi and Ho Chi Minh City, but the only scheduled transpacific flights are to Honolulu and Guam.
The airport is split into 4 terminals.
T1 and T2 are essentially different parts of the same building, 5 min apart on foot. The subway station is located under Terminal 2.
From Tokyo, flying to Fukuoka is much faster than the Shinkansen, and not significantly more expensive. The usual one-way fare on Skymark Airlines from Tokyo Haneda is ¥19,800, compared to ¥22,320 from Tokyo Station on the Nozomi Shinkansen, and steep discounts are available if you book in advance (as low as ¥5,300 with Skymark's "SKY Bargain" discount). The flight takes 2 hr while the train takes five. If you have a Japan Rail Pass (JR Pass), of course, you'll still want to take the train, though you can't take the fastest of the Shinkansen ("Nozomi") with the JR Pass.
Fukuoka's Hakata Station is the terminus of the Sanyo Shinkansen (north) and Kyushu Shinkansen (south). Sanyo Shinkansen services are offered from Kokura in Kitakyushu (20 min), Hiroshima (1 hr), Okayama (1 3/4 hrs) and Osaka (2 hr 30 min), and through via the Tokaido Shinkansen from Kyoto (2 hr 45 min by Nozomi), Nagoya (3 hr 30 min by Nozomi) and Tokyo (5 hr by Nozomi).
If you have a Japan Rail Pass, you cannot use the Nozomi, so if you are traveling from Tokyo or Nagoya you will have to take one of the two hourly Hikari trains from Tokyo and change at Shin-Osaka (alternatives are Shin-Kobe, Okayama, and sometimes Himeji) to a Sakura service. Travel time from Tokyo to Fukuoka using these trains is 6 hours.
Another option from Tokyo is to take a westbound sleeper express such as the Sunrise Izumo or Sunrise Seto, leaving Tokyo around 10PM, and then connecting to the Shinkansen at Okayama early in the morning, to arrive in Fukuoka just before 8:30AM (or by 9:15AM if you have a Rail Pass and use a Sakura service). While this takes much longer and costs more than the Shinkansen (from ¥25,000), it provides the benefit of doubling as lodging and transport.
From Kagoshima, Kyushu Shinkansen Mizuho and Sakura trains make the run to Fukuoka in 80–90 minutes at a cost of ¥10,170. The Mizuho is NOT valid with the Japan Rail Pass. Most Sakura trains do travel through Fukuoka, connecting Kagoshima to Osaka with no transfers.
From Nagasaki, the limited express Kamome runs hourly (sometimes twice an hour), taking 2 hr and costing ¥4,710 each way.
For historical reasons, Fukuoka's train station is called Hakata. If you search for schedules to "Fukuoka" online, you will likely be given an itinerary for a totally different (and much less interesting) city in northern Japan.
If you hold a Japan Rail Pass, and you wish to travel overnight from Tokyo (or any other distant city), you may want to split up your journey, stopping at an intermediate destination en route in order to sleep somewhere. The cost incurred will only be for the hotel room; the Rail Pass covers your transportation. This is a good way to travel overnight, especially if you are able to find cheap accommodations, such as a business hotel. Yes, it may be a little hectic, and it might require some research, but this method carries two significant advantages: location and money. You will more than likely find good accommodations very close to a main train station in a smaller city, compared to a big city such as Tokyo, and it will more than likely be cheaper than hotels found in big cities. You could use the money you save to forward some of your luggage to Fukuoka using a luggage delivery service and take an overnight bag with you, which will make the journey easier.
As of March 2012, here is one way you could go about this from Tokyo: at 7PM, leave for Himeji by taking the Hikari train and changing to a Kodama service at Shin-Osaka station. Once in Himeji (arriving around 11PM) you can take a rest at Himeji's Toyoko Inn which costs as low as ¥5230 for a single room or ¥2990 double occupancy. At 6:54AM the next morning, board the first bullet train of the day, a Sakura service, and you will be in Fukuoka by 9:15AM. This trip takes longer than taking the overnight train and bullet train connection as described above, but it is cheaper; you only have to pay for the hotel room, complete with your own toilet and shower.
Many overnight bus services run into Fukuoka from other parts of the country.
The Moonlight overnight bus runs from Osaka Umeda to Fukuoka in 9 hr 30 min (¥10,000 one way); The Kyoto overnight bus runs from Kyoto to Fukuoka, also in 9 hr 30 min (¥10,500 one way); and the oddly-named Dontaku runs from Nagoya to Fukuoka in 11 hr (¥10,500 one way).
Willer Express has a service from Osaka/Kyoto/Kobe from ¥4,800 with advance purchase tickets as cheap as ¥4,100. Other services are Nagoya (¥5,400), Okayama (¥4,300) and Hiroshima (¥2,500). They have an English website with online booking available. Discounts for tickets purchased 21 and 14 days in advance.
If you're really ambitious, Nishitetsu bus runs an overnight service, the Hakata, from the Shinjuku expressway bus terminal in Tokyo to Fukuoka non-stop. The ride, at just over 14 hr, is Japan's longest overnight bus service (¥8,000 for economy class ¥12,000 for high seasons, ¥15,000 for business class and ¥19,000 for 1st class, some roundtrip discounts are available).
JR Kyushu's Jet Ferry the Beetle hydrofoils to Busan (South Korea). It runs five times a day and takes just under 3 hr for ¥13,000 (¥24,000 round trip discount fare; ¥20,000 round trip on weekdays). They are quick, but in 2005 one hit a whale and had to be towed back to Busan. Since then, the JR Kyushu Jet Ferry Inc. plays sounds that whales dislike using speakers to avoid further accidents. An economy-class ticket on the Meimon Taiyo Ferry from Osaka to Kita Kyushu costs ¥6,000 (20% discount if booked online); tickets in other price ranges are available.
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