This feels more fighter-ish than barbarian-ish to me to be honest (as for that matter is Conan; how often does he rage?) I fail to see how for an anti-magic approach wrapping yourself in cold iron isn't a good way to start (unlike raging). Of course the Shadow Monk can also be pretty great as an anti-magic subclass.
But ultimately I think that this sort of thing is what feats (like Mage Slayer and possibly a couple of others) are for.
I guess I'm not viewing at it as "beating magic with anti-magic." (That was probably a bad word-choice on my part!) It seems to me as if Conan mostly overcomes magic with the force of "good ol' fashioned steel, thews, and grit!" Counterspells and such seem pretty out of character for him. In my limited acquaintance with the stories, I don't recall him using magic to beat other magic at all. It always always boils down to one or more of of: Punch the fiend! Stab the caster! or Smash the macguffin!
As I recall, Conan doesn't typically wear enchanted armor or use special equipment* or anything. And he doesn't seem to understand magic, so much as he just greatly distrusts it. He usually wins just by swallowing his fear, clenching his teeth, and punching the monster til it stops twitching. And more often than not, his supernatural encounters involve some degree of intuition on his part to discern a weakness or pierce a deception or the like.
Anyway, that's why I'm envisioning an archetype based on resisting magical effects and hurting magical beings with fists, rather than focusing on dispelling and the like. Though to be fair, Conan
did stab more than a few evil sorcs in the face, so the Mage Slayer feat certainly fits!
(edit: To be clear here, I'm not trying to model the full character of Conan, just this one facet of the character.)
Anyway, that's just my 2 cp!
* Though I suppose his furry underpants in the early comics could have been magical.