Hi there,
Fukuoka is a fairly quiet airport, so I'd predict you'll be through customs within an hour of landing. The airport is only 6 mins from Hakata Station on the subway (sample itinerary), and the JR ticket offices at Hakata are open for activating JR Passes right up until 23:00 so you should be able to get to the platform ready to board a bullet within 2 hours of landing. You are correct that you'll need to change trains en route up to Nagoya and will not be able to use the Nozomi or Mizuho class bullets.
I've just checked the timetables, and it seems the last valid shinkansen bullet up to Nagoya leaves Hakata at 18:38, which will be too late for you. The schedule could be different on the day you choose to travel but it's worth double checking before you finalise your plans (I checked the trains for today and tomorrow, both had the last trains with connections to Nagoya leaving at 18:38, getting you into Nagoya at 22:35).
If you do stay a night in Fukuoka, I highly recommend the Plaza Hotel Tenjin (& tripadvisor) for it's good location (daimyo), style and price.
Hope this helps!