Where to stay depends a lot on your budget. But areas with lots of hotels are near Hakata Station, the Gion area, and near Tenjin. You can see a good collection on cheaper places as well as sightseeing info on:
http://www.thejapanfaq.com/fukpic.htmlAs you have a whole day, regarding what to see, I'd recommend you see the best temple in the whole area - and nearly totally unknown to the tourist guides: about 20 minutes by train from Hakata Stn is Nanzoin, which blows away many more famous places in Japan in terms of size and sheer beauty:
https://www.youtube.com/watch?v=lT4VQFmHFyMFor food, I'd also say that trying the local cuisine is important - and in answer to your question aside from the tonkotsu ramen, mentaiko (spiced salmon roe) is a big favorite. So are other dishes like mizutaki and motsunabe, but eaten more in winter. You can see a list of places to eat on
http://fukuoka-now.com/food/archive/but I wouldn't waste too much time in searching for a restaurant.
For art you might be interested in the Fukuoka Asian Art Museum, and Fukuoka Prefectural Art Museum. There is also the Kyushu National Museum, but that is in Dazaifu. Aside from that you might like the Hakuhaku, a museum of Hakata food and culture, which is also next to a mentaiko factory where you can do some tasting, plus the Hakata Machiya Folk Museum.
For yatai most people go to the Nakasu area, which along with the neon lights on the river make a nice scenery - but many locals go to the Nagahama area near the waterfront. Most yatai start getting set up in late afternoon and are ready to go just before dinnertime.
Lastly, regarding gardens, you might like Yusentei and Rakusuien. The former is an old villa and a bit harder to get to; the latter is next to Sumiyoshi Shrine and easy to reach.
Both are very inexpensive. The best garden though would be the Nokonoshima Island Park - quite out there and pricey but a beautiful place with a grand view over Hakata Bay:
https://www.youtube.com/watch?v=H-RmFcmhs9cMany places you can easily reach by subway, which is comparatively expensive to other cities, so a subway day pass might help. Or if you want to add buses, look into a Fukuoka Tourist City Pass:
http://goo.gl/XFQSwe