UPS allows shipping via different next day air services:
Handguns, as defined by 18 U.S.C. § 921, will be accepted for transportation only via UPS Next Day Air Services, specifically, UPS Next Day Air® Early A.M.®, UPS Next Day Air®, and UPS Next Day Air Saver®. (Note: UPS Express CriticalSM Service is not available for firearms).
I think the Air Saver is the least expensive (note that didn't say "cheapest").