Last time I went through immigration in FUK was in December 2022, when I arrived on a Saturday at 7:35pm from ICN on t'way. Considering how small the airport is, I was surprised that it wasn't the breeze I expected. Probably spent a good 20min to pass through immigration, and another 5-10min at customs. (From what I remember, they also initially only had one counter open for international visitors, but opened more some minutes after all of the passengers from my flight had arrived at immigration.)

Also very important (!): The subway station is at the domestic terminal, not the international one. So even though Hakata Station is just minutes from FUK by subway, from the international terminal, you might be quicker taking a taxi if you're in a hurry.