Kitakyushu area is just over there to Yamaguchi and Shimonoseki on the mainland.
The Shinkansen, Nozomi, stops at Kokura Station.
Kitakyushu Airport is the closest airport to Kokura, with flights from Haneda.
It takes about 45 minutes to an hour by limousine bus from the airport to Kokura.
From Fukuoka Airport, if you take the Shinkansen between Hakata and Kokura, it takes roughly 15 minutes from Hakata to Kokura, but if you consider the transfer from the airport, it takes about 40 minutes, which is not much different from arriving at Kitakyushu Airport.