Originally Posted by msb0b

Even on Narita Express + Shinkansen you would be stuck on the train for 7 hours vs 2 hours on a plane.

The flight options from NRT to FUK after your arrival time are (T2 indicates departure from Narita Terminal 2):
15:25 JL3053 (T2)
15:30 BC805 (T2)
17:00 NH2143
17:55 NH2145
18:00 GK127 (T2)
19:50 JL3057 (T2)

But if you insist on taking the train, and assuming you will use Japan Rail Pass, the latest train connection that will get you to Hakata is:

Narita Terminal 1
| 15:45 Narita Express 34 Arr 16:52
Shinagawa
| 17:10 Hikari 481 Arr 20:18
Shin-Kobe
| 20:53 Hikari 443 Arr 23:52
Hakata

A bit more forgiving in reverse but you still need to catch the train by 8:00 to get to Narita by 15:30.