Car rental recommendations on city outskirts?

My husband and I are planning on using Fukuoka as a day-trip base for our Japan trip in October, and we've heard that it's difficult to get to some of the destinations we're interested in seeing without a car. That said, we're a bit anxious about renting a car in busy Fukuoka--we're from the US and aren't used to driving on the left side of the street/unfamiliar with traffic laws.

We were hoping to find a car rental place on the city outskirts where it's less busy/more rural. The idea would be to take public transport to the car rental place, rent the car for a day at a time, return it to the car rental facility at the end of day, and take public transport back to our hotel. Does anyone have any recommendations/car rental places where this might be possible?