If you do not have an ffl (and maybe a C&R) license, don't try it. To make things tougher, current antique definition at the Post Office is only single shot muzzle loading firearms according to the references below. Some extracts from the USPS are below, but as DWalt states, still quite confusing.
12.1.3 Authorized Persons
Subject to 12.1.4, handguns may be mailed by a licensed manufacturer of firearms, a licensed dealer of firearms, a licensed importer of firearms, or an authorized agent of the federal government or the government of a state, territory, or district.
12.1.5 Manufacturers, Dealers, and Importers
Handguns may also be mailed between licensed manufacturers of firearms, licensed dealers of firearms, and licensed importers of firearms in customary trade shipments, or for repairing or replacing parts.
12.1.2 Handguns
Handguns and other firearms capable of being concealed on the person are nonmailable unless mailed between the parties listed in 12.1.3 and 12.1.5 after the filing of an affidavit or statement required by 12.1.4 and 12.1.6, and are subject to the following:
a. Firearms meeting the definition of a handgun under 12.1.1b, and the definition of curios or relics under 27 CFR 478.11 may be mailed between curio and relic collectors only when those firearms also meet the definition of an antique firearm under 12.1.1g.
b. Firearms meeting the definition of a handgun under 12.1.1b that are certified by the curator of a municipal, state, or federal museum that exhibits firearms to be curios or relics of museum interest may be accepted for mailing without regard to 12.1.3 through 12.1.6.
c. Air guns that do not fall within the definition of firearm under 12.1.1a that are capable of being concealed on a person are mailable; but must include adult signature service under 503.8.0. Mailers must comply with all applicable state and local regulations.
d. Parts of handguns are mailable, except for handgun frames, receivers, or other parts or components regulated under Chapter 44, Title 18, U.S.C.
Definition: Antique firearm means any muzzle loading rifle/shotgun/pistol that is designed to use black powder or a black powder substitute and that cannot use fixed ammunition (except those that incorporate a firearm frame or receiver, any firearm that is converted into a muzzle loading weapon, or any muzzle loading weapon that can be readily converted to fire fixed ammunition by replacing the barrel, bolt, breechblock, or any combination thereof); or any firearm (including those with a matchlock, flintlock, percussion cap, or similar type of ignition system) manufactured on or before 1898, or any replica thereof, if such replica