Cycle in Fukuoka

We are planning to visit Fukuoka mid January 2016. Is it advisable to cycle in and around Fukuoka?

You certainly can, though rates vary widely and the winds can be bitterly cold on a bike at that time. You can get more info at:
http://fukuoka-now.com/en/2014/11/rental-bicycle-guide-for-fukuoka/
Alternatively many of the best sights are close to a subway station, so another option would be to get a subway day pass or together with the bus there is the Fukuoka Tourist City Pass:
http://goo.gl/XFQSwe
One of the very best sights though you'd need a 20 minute train ride from Hakata Station - the Nanzoin Temple is fabulous and you'll never forget it.
https://www.youtube.com/watch?v=lT4VQFmHFyM
For other ideas on what to see and do, have a look at:
http://www.thejapanfaq.com/fukpic.html
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...

Thank you very much for the valuable information that you have shared with me. Really appreciate it very much Ken.