Yes, it is possible if you are given a CITES export permit from Russia for the sheep before you leave and follow the USFWS and USDA rules for import. I take an extra waterproof duffel. The skin must be salted a few times before you leave so it is fairly dry. The skull and horns must be clean and soft tissue removed. I brought my Marco Polo ram home from Tajikistan this way, my Stone’s Sheep from BC and my desert bighorn from Mexico. I have also accompanied several desert bighorn hunters and their rams back to the USA when I used to guide down in Baja Sur.
Make sure you have a filled out USFWS Form 3-177 with you, the original CITES export permit (this is kept by USFWS but I like to keep a copy) and follow the USDA rules for importing skins, skulls and horns. Find the rules online on their websites and print them. I have often found that I know more about importing sheep horns, skulls and skins than the CBP officers, USFWS officers and USDA inspectors than they do and it is helpful to have the rules printed out to show them. It is also important to stay calm and act confident as they figure things out. Make sure you have plenty of layover time between flights and that your arrival US city’s airport is an approved USFWS port of entry. Not all airports have USFWS port of entry designation but most major airports do have this designation.
When I would fly in to LAX after the Mexico sheep hunts, I called ahead to a USFWS Officer/Inspector at the Torrance, CA office and he would meet us at Customs at LAX. If he couldn’t make it, he had me give the CITES export permit in a stamped and addressed envelope to a CBP Officer, who would then mail it to him. You also need a copy of the hunting contract from your outfitter.
Let me know if I can be of further assistance. Be sure to check the rules to see if anything has changed since 2018, the last time I imported a sheep.