Have lived on Saipan for 1 year now. Have never seen any lockers or luggage check service at the airport. TBH, for that short period of time, I would just not pick up my luggage, go explore the island and then claim your luggage when you get back to the airport. The ticket counters are not open when there are no flights scheduled, they open every few hours. So you can't "check in" with your bags when you arrive.

Best bet is to rent a car (you'd need it to get around anyhow) and put the bags in the car. The rental car center is right at the airport.