There are taxis in the port waiting for regularly scheduled ferries so I guess there would be some for a cruise ship, but I've never arrived on a cruise ship so I don't know for sure. There are buses, but your first mode is best as taxi to get you to a subway or train station. The main subway runs from the west side of the city to the airport via Tenjin and the Hakata Train Station. Places aren't that far in Fukuoka, so with 4 people you would save time and not spend much more by using taxis. The exception would be to get to Dazaifu, where a train would be better. See if they hand out maps on board with locations in Japanese before you disembark so you can show the taxi drivers where you want to go. Also, get a copy of the port address of your cruise ship in Japanese so you can quickly get back. Having your destinations written out in Japanese saves the taxi drivers stress and gets you to your destination quickly without mistakes.
What you want to see and do varies by person, so I can only give some suggestions.
If you like crafts, there's the Takatori Pottery on the west side Ohori Park.
http://takatoriyaki.com/en/overviewThe city museum often has interesting special exhibits and isn't too bad for the history of Fukuoka.
http://museum.city.fukuoka.jp/en/Canal City is a shopping attraction, although I find the architecture more interesting than the shopping.
http://canalcity.co.jp/static/detail/visitors_enThe Acros Building in Fukuoka is interesting:
http://www.travellersbazaar.com/green-architecture-the-acros-fukuoka-b...General list of places to go:
https://kyushu.com/fukuoka/places_to_go/folk_museum/1,museums_gallerie...There's not much left of the castle ruins and Ohori Park, based on Xifu (West Lake) in Hangzhou, is nice for a walk but not if you're trying to cut down on walking.
There are also remnants of the walls built by the Japanese in the 13th C to repel the Mongols, but most you have to walk a ways to. There's some remnants near the City Museum.
Dazaifu can be of interest, but the old government buildings can just about be totally seen in the photo on this page:
http://www.japan-guide.com/e/e4850.htmlThere are just foundation stones left. The new museum there would make visiting the stones have a bit more meaning.
To get there, you'd leave from the Nishitetsu Station in Tenjin; it's easier than using the JR trains.
http://www.japan-guide.com/e/e4854.htmlWhen I pull into Hakata Station, I usually check with the tourist info desk in the middle to see what's going on in town. You can pick up city maps here as well. There's also a tourist info center in the Tenjin Nishitetsu Station if you plan on Dazaifu.
More info:
https://en.wikivoyage.org/wiki/FukuokaFukuoka used to be the west side of town with the castle and Hakata the working man's city on the east bank. The station was built in the east section so it's called Hakata Station. The airport is newer and built after the 2 cities joined under the name Fukuoka so it's called the Fukuoka Airport.
It might be a bit early for a pint but I stop in at Mike's Pub, now called the Craic and Porter on Oyafuku-dori in Tenjin.
