Unless you just don't like it, trying some of the Hakata ramen is worth a go, not to mention some mentaiko (spiced cod roe). If you want to try the yatai or ramen stalls on the street, you'll mainly find them along the Naka River in Nakasu from the evening (best scenic area as well, by all the lights). You can also find some in Tenjin, which is the main shopping district.
If you don't want or can't do that, you can go to the Nagahama area near Tenjin - there are some brick and mortar places for it as well as yatai, or at Canal City (5th floor) in between Tenjin & Hakata Station, there is Ramen Stadium, which is a bunch of small restaurants with a variety of different noodles.
As for what to see and do, it depends on your interests, but one of the best sights is just outside the city, the Nanzoin Temple. It has become a lot more well known over the last decade, but is still a hidden gem to many. It's large reclining Great Buddha by itself is worth the effort to see.
https://youtu.be/lT4VQFmHFyMAlso lots of shopping if you like, just before leaving Japan. There is Hakata City around Hakata Station, Canal City, Tenjin, plus around the Paypay Dome (TeamLab Forest is there too). You could also see Tochoji next to the Gion subway station, for the country's largest carved wooden Great Buddha.
Ohori Park is one of the nicer and memorable parks in the city - if you go, don't miss the nice traditional garden on the park's south side. Maizuru Park is next door with the castle ruins and some nice quiet areas, plus Korokan, which is partly an archeologic site where visitors from Asia used to visit and be held, is there as well. Nishi Koen is a nice park about 10 minutes on foot away from there too, with lots of hillside walks. If the weather is good and you want a nice view, there is also Fukuoka Tower in Momochi. Lastly, as mentioned, there is Nakasu, which is no Shibuya by any stretch, but still work going through.
https://goo.gl/X3LDaE Lots more of course, but if you have less than a full day and you need to catch a flight early the next morning, limiting your time in transport makes more sense.