I'm afraid I don't understand the question. There is only one train route by bullet train, taking a minimum of 5 hours on Nozomi trains (which you cannot ride with the full JR Pass). If you use the pass, it would add another 40 minutes and in almost every case you'd need to change trains at least once. The most efficient way would be to fly from Haneda Airport to Tokyo - you can get there in about 90 minutes, Fukuoka Airport is one of the lucky few in Japan actually inside the city, and there are dirt cheap airfares like ANA's Experience Japan Fare, or on low cost carriers like Jet Star or Peach (though they leave from Narita). http://ana.ms/20Xtf8t http://www.jetstar.com/jp/en/home http://www.flypeach.com/pc/en
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The most efficient way would be to fly from Haneda Airport to Tokyo - you can get there in about 90 minutes, Fukuoka Airport is one of the lucky few in Japan actually inside the city, and there are dirt cheap airfares like ANA's Experience Japan Fare, or on low cost carriers like Jet Star or Peach (though they leave from Narita).
http://ana.ms/20Xtf8t
http://www.jetstar.com/jp/en/home
http://www.flypeach.com/pc/en