Short trip in Fukuoka? There is Dazai Tenmangu and Toto factory to visit
In Kyushu, there are many lovely things to visit and usually, Fukuoka, is not a must-go on the list.
But, if you go there, you can certainly enjoy your time.
Dazai Tenmangu is a temple dating from the 10th century and built by a Michizane Sugawara, an high-rank administrator of the Heian government who was demoted. So he left Heian and went to Fukuoka where he lived in exile and poverty. But he was a scholar, so goes the history.
So today, Dazai Tenmangu, a shinto shrine, is the place to go if you want to pray for your academic success. It is also a very beautiful place in the mountain, 45 minutes by bus from Hakata station bus terminal.
It was pouring cats and dogs when I went but I enjoyed in very much nevertheless. There are a lot of "hortensia - ajisai" and huge plum trees.
I ale tasted the local ume-manju, baked in front of you. The inside is your usual anko, sweet bean paste (though it is one of the best I have ever eaten) and the surrounding is made of kochi-gome, sticky rice. This alone is worth the trip.
Kokura is the next city on the shinkansen, from Fukuoka. It is closer to Honshu than Fukuoka. Actually, from the historical city, colonial looking place, you can see Honshu on the other side of the sea. Kokura is a small town, and walking around it is very enjoyable but if you go, there is one place you absolutely need to visit: Toto factory.
Yes! Kokura is also the hometown of Toto, the toilet manufacturer. If you want to know everything about Japanese toilets, you will want to visit this ceramic factory. It is awesome.