Please be aware that Tokyo has multiple airports and that these results are for all airports serving the city.
The best month to fly between Tokyo and Fukuoka is February and the most expensive month to fly is October. (Average prices, based on 1223 datapoints.)
The cheapest airline flying from Tokyo to Fukuoka is All Nippon Airways. They are 8% cheaper than ANA Wings. (Average prices, based on 1254 datapoints.)
The best time to fly is at around 16. (Average prices, based on 1285 datapoints.)