Young Dolph is recovering from surgery after getting shot in Hollywood, but he won't be sitting down, comfortably anyway, for a while ... TMZ has learned.

Sources close to the Memphis rapper tell us he was hit at least twice -- probably more -- during the shooting. We know he was hit in one of his arms and his butt. He underwent surgery Tuesday night, and we're told he'll be okay -- his injuries are non-life threatening.

As we reported ... LAPD has arrested one person in connection with the shooting, and our law enforcement sources tell us Dolph's rival, Yo Gotti, is a person of interest.