Fukuoka undoes the usual arithmetic of choosing a hotel. In most cities the question is how far your bed sits from the things you came for, and the answer carries a price. Here the subway from the airport reaches Hakata Station in about five minutes and Tenjin in around eleven, because Fukuoka Airport sits inside the city rather than beside it. Hakata and Tenjin are themselves roughly six minutes apart on that same Kuko line, and a taxi across the middle of town is a short ride. Nothing is far from anything, and no district can sell you a convenience you would not have anyway.
So the distance question answers itself, and a more useful one takes its place: what is outside your door late at night, and what is outside it early the next morning? Fukuoka eats outdoors. The yatai - folding stalls assembled on pavements and riverbanks at dusk and packed away before dawn - generally begin serving in the early evening and keep going into the small hours, which is well past the last train. Where you sleep in this city decides whether the night ends with a slow walk along the water or a taxi you have to negotiate in gestures.
First, stop worrying about the airport
Fukuoka Airport is joined to the city by the Kuko subway line, with a station directly beneath the domestic terminal. Two stops gets you to Hakata; a few more carry you through Nakasu-Kawabata and Tenjin and on westward. Fares sit in the ordinary Fukuoka city-subway range - a few hundred yen at most for anything inside the centre - and the major IC cards all work, including Suica, PASMO and ICOCA alongside the local SUGOCA, nimoca and Hayakaken. Paper tickets are still sold from the machines beside the fare chart.
The practical consequence is that Fukuoka has no airport-hotel district and no reason to want one. A late arrival can be in a bed in the centre within half an hour of clearing the gate, provided the subway is still running, and an early departure does not demand the usual pre-dawn contingency planning. It is also the reason Fukuoka deserves two nights on a Kyushu itinerary that might otherwise have given it one: the hours you would normally lose to airport transfers simply are not spent.