I must have missed you mentioning what time you are thinking of traveling.
During Golden Week (1st week of May) the wisteria at Kawachi Fujien are stunning. But so are the crowds, and it is a pain to get to. You can't go there and just buy a ticket - with so many people there now you have to get one at a convenience store. Search the forum for past answers on how to get to there. If not in Golden Week, most of the time the place is not open.
Regarding the places to see in Fukuoka listed above: the Acros Bldg is only really worth it on the weekends when you can climb it to the roof. Nobody told me about the weekends part when I last went, which resulted is a lot of trouble. Hakozaki Shrine (not Temple) is quite historical (it was burned down in the first Mongol invasion in 1274) but one of the real gems is its flower garden for peonies and hydrangea (May & June mostly). Tenjin Central Park has little and not worth much attention. Gokoku Jinja is worth a brief look next to Ohori Park, but nothing worth going out of your way for.
Some other nice places are Nokonoshima for its hiking and Island Park, (especially cherry blossoms, canola and in the fall, cosmos), there are some nice traditional gardens at Rakusuien, Yusentei, and Ohori Park, plus the views from the Fukuoka Tower or Minami Park Tower are excellent when the weather is good. The Gion area has some nice temples - Shofukuji as mentioned plus Japan's biggest carved wooden Great Buddha (about 11m tall) at Tochoji - and it's free. There are tons of small temples in the area but most aren't open to the public.
Nakasu is the night time district and try out some Hakata ramen at one of the ramen stalls (yatai)
along the Naka River there. The smell is quite pungent but they are very proud of their ramen.http://www.thejapanfaq.com/fukpic.htmlhttp://bit.ly/1KAm8cU
There is also a new 2 day subway pass you can use
and many places are close to one of the subway stations.
Some other day trip possibilities are Karatsu in Saga, Kitakyushu (Kokura Castle, Mojiko, Sarakurayama), Shimonoseki (Kanmon undersea tunnel, Hi no Yama, Kaikyokan aquarium, Akama Shrine etc).
Tenjin is the main shopping area, and you can find some good watches at the large Bic Camera there, or near Hakata Station there is a Yodobashi Camera shop. Maybe at Canal City but that is more clothing and boutique shops. If you're from Singapore, you'll probably find something cheaper where you are though.