I generally understood a Mary Sue to be a perfect character written into a story (like an episode of a TV series, for example), who was able to show up all the other established characters who came before them. From that aspect, Conan certainly isn’t. But if a female barbarian suddenly got dropped into the Conan universe, and was stronger/smarter than even Conan, she would be an example of a Mary Sue.
Rey from the Star Wars sequels is, I think, a better example of a Mary Sue.
Nowadays it seems that the idea of the Mary Sue has somehow been stretched to mean any hero who is perfect/flawless. From that standpoint, Conan might fit the bill if we assume he is REH’s idealized vision of Man. He wouldn’t be if we judged him by the so-called standards of the so-called modern audiences of today.
Hither came Conan the Cimmerian, black-haired, sullen-eyed, sword in hand, a thief, a reaver, a slayer, with gigantic melancholies and gigantic mirth, to tread the jeweled thrones of the Earth under his sandalled feet.
I think it comes down to which version of Conan you are discussing.
If we stick to the original Howard stories, Conan is flawed and isn't a do-everything type. Conan gets tricked, he gets knocked out, he shows occasional fear or weakness, he is a "real" person. I continually go back and re-read those stories.
If you include a lot of the extra pastiche novels I think the character grows into a superman type which could be classified as a "Mary Sue" because Conan almost evolves into a parody of himself (me barbarian, strong, distrust magic). I quit reading the novels for that reason.
I have a hard time with folks who try to stat out Conan for RPGs like D&D. They end up giving him all 18's or higher because he is super strong, tireless, lightning quick, and so on. I see Conan as the ultimate in Strength but El Borak seems faster. The RPG version of Conan seems to match the pastiche Conan more than the Howard one, and thus is probably a "Mary Sue" as well.
The Schwarzenegger movie Conan seems pretty human, and isn't a do-it-all type of character. He shows a lot of strength but not as much speed. He can sneak but isn't a master thief. He can fight but isn't the greatest swordsman ever. He is surrounded by other capable characters who can do the same things he can do and (other than feats of strength) often can do them better.
So I suppose the only way to really discuss this is to define which version of Conan is under consideration.
As mentioned in Von K's post above it's well worth checking out Taranaich's post over at The Blog That Time Forgot Anyways, here's the conclusion of that post:
Conan is and isn't a Mary Sue, depending on what you think of Conan, and which Conan you're talking about. Howard's Conan, however, most certainly is not a Mary Sue.
Conan fails as often as he succeeds. He's unsuccessful in attaining the treasure he set out for in "The Tower of the Elephant," "The God in the Bowl" and "The Servants of Bit-Yakin." "Beyond the Black River" ends on a considerable downer, while "The Phoenix on the Sword" has a sobering and unsettling denouement. "Xuthal of the Dusk" ends with Conan literally flayed alive, and in more than a few stories he runs in stark terror from the Eldritch Horror instead of slaying it. There are some where he would flee, such as "The Devil in Iron," but when the girl-of-the-week's in danger, he overcomes the fear. What's more, he comes very close to defeat and death in some battles, with only luck or outside interference saving his skin. Even mere mortals like Ascalante, Aratus and an unnamed Nemedian soldier have come close to ending the saga of Conan.
What about everybody loving/jealous/wanting to be him? That would certainly make his reign as king a heck of a lot easier if he were universally adored, rather than the subject of assassination attempts, baronial revolts and invasions. Every "informed ability" is explained as something he learned from Zamorian philosophers, Pelishti wise men, or Nemedian scribes, and his multiple languages are a result of a keen mind eager to learn in a world where such skills are common. There are plenty of amazing people in history like T E Lawrence, Teddy Roosevelt and Richard Francis Burton who show similar capacity for intellectual subjects combined with great physicality: Conan's just a mythologised version of that breed of man.
As for "author insert", Conan, from Howard's own admission, is based on people he knew. So the man himself says that Conan isn't merely an idealized author insert. Perhaps on some very basic level, maybe, but it isn't as clear cut as many seem to make out. Conan is not invulnerable, not everybody worships him, not all girls fall at his feet, and he's not an invincible juggernaut in combat.