One does not need to go a few days in advance.
Helping a friend sell an inherited collection we mailed a lot of guns he sold via USPS. Just package it securely, do NOT have gun or similar on the label.
Example, if the FFL recipient is Joe Bob at Joe Bob's guns,
I address the package to the buyer, c/o Joe Bob, address, state, zip - no mention of the business name
Get a plain brown shipping box from a gun dealer, and make sure - as stated above - that it is secured so the barrel does not come poking out.
One way to do that is to buy a cheap plastic gun case and get the brown box it came in