Best ways from Fukuoka Airport to Dazaifu

Ask for help again. If we arrive at Fukuoka airport at 9 o clock, can we make a trip to Dazaifu? We have to board a cruise at 2pm. Thinking of leave luggages at the airport and go there. And back to get luggage and go to the Hakata port. Or should I bring all luggages along.

Absolutely no need to bring your luggage along.
The easiest way is by Nishitetsu train from Tenjin Station, with one transfer at Futsukaichi Stn.

If you have the time and desire, Futsukaichi has several hot springs, some going back over 1300 years.

You can see a preview of the Dazaifu Shrine at
https://www.youtube.com/watch?v=BwxGPZxPlNw

There isn't gonna be a 100% perfect way to do it. But I'd agree with the above. Take subway to Tenjin. Put bags there. Take Nishitetsu train to Dazaifu (there will be another transfer). Return to Tenjin, then Taxi to port.

There are other options like direct buses from Airport and JR trains from Hakata station, (but this option will require a few more transfers).

I just re read your question. Does you plane get in a 9am? OR is that the estimated time to exit the airport?

One thing to note is that from the airport to Dazaifu is about 60 minutes travel time with transfers. This 60 minutes doesn't include finding a luggage locker, buying tickets, having lunch etc.

You have 5 hours, so removing 90 minutes each side of Dazaifu will leave you about 3 hours spare at a miximum. Less if you flight actually arrives at 9am. I would have a back up plane for late arrival etc. there is some nice shopping and smaller temples along the subway line if you have limited time.