From Fukuoka International Airport to Hotel AZ Itoshima
There are two ways to come.
A. From International Terminal => Hotel AZ Itoshima (Taxi)
B. Use Shuttle bus, Subway & JR train, and Taxi or Walk.
B-1. International Terminal => Domestic Terminal (Shuttle Bus)
For further information, see Fukuoka Airport Shuttle Bus Route Map
B-2. Fukuoka Airport Station => Hatae (波多江) Station (Subway & JR Train)
Please get on the train bound for "Chikuzen-Maebaru(筑前前原)" or "Nishi-Karatsu(西唐津)" NOT to "Meihohama(姪浜)". Then, you can get to Hatae Station directly without changing train.
For further information, please see Fukuoka City Subway Route Map
B-3. Hatae Station => Hotel AZ Itoshima (Taxi or Walk)
Most convenient (and therefore recommended) way is to use taxi.
If you walk there, it may takes about 30 minutes.