FFL holders are allowed to mail a handgun to another FFL holder via the U.S. Post Office. That is a lot cheaper than UPS or FEDEX.
It is a little more complicated than that because not all FFLs can use the post office - a 01 license holder can, a 03 license holder (C&R) cannot.
But the simple answer is that if a handgun is going from a dealer to another dealer the mail can be used.
Now, as regards UPS policy, here is the UPS policy from their own site:
"Handguns, as defined by 18 U.S.C. § 921, will be accepted for transportation only via UPS Next Day Air® Early A.M.®, UPS Next Day Air®, or UPS Next Day Air Saver® services."
http://www.ups.com/content/us/en/resources/prepare/guidelines/firearms.html
The "next day air" thing is an internal UPS policy. I read once that it was due to UPS concerns about handguns being stolen from their system because of the delays inherent in ground transportation. I don't know if that is correct.
If the guy you bought from shipped a handgun via UPS ground, that is between him and UPS to resolve.