Assume we are given an annulus
$$A = \{ z \in \mathbb{C}: 1< |z| < R\}.$$


Let $\phi\colon A \to A$ be a univalent map such that the image of $\phi$ contains a curve around the unit disk. Does this imply that $\phi$ has to be in fact an automorphism of the annulus?



Can an "annular" subset of an annulus be conformally equivalent to the whole annulus?
The answer to the question in your last sentence is yes, and this follows from the Schwarz lemma, which says that a holomorphic map between hyperbolic Riemann surfaces strictly compresses the hyperbolic metric, unless this map is a covering. Equip your annulus $A$ with the hyperbolic metric, and let
$\gamma$ be the shortest curve in the class of curves separating the boundary components. If your map is not surjective then
it must strictly compress the hyperbolic metric, so $\phi(\gamma)$ is strictly shorter than $\gamma$, which is the contradiction. 


