For any antiholomorphic Diffeomorphism $f\colon S\to S$ we get a canonical identification $f^\star\bar K=K,$ $ K $ and $\bar K$ being the canonical and anticanonical bundle of the Riemann surface. A holomorphic structure on a complex vectorbundle $E$ is the same as an complex operator $D\colon\Gamma(E)\to\Gamma(\bar KE)$ satisfying the (Cauchy Riemann) Leibnitz rule. (holomorphic sections are exactly the one in the kernel of D). (For higher dimensions this is not true.)
Now, $f^* E$ has a natural complex structure (it's just i). Therefore one gets an anti-holomorphic structure $\bar D\colon\Gamma(f^\star E)\to\Gamma(Kf^* E)$ satisfying the antiholomorphic Cauchy Riemann equation. But the complex conjugate bundle $\bar E$ also has a anti-holomorphic structure, since $\overline{\bar K E}=K\bar E.$ Therefore, $f^* \bar E$ has a natural holomorphic structure.
In the case of a line bundle $L=E$ this holomorphic structure on $f^* L$ is isomorphic to the one on $L.$ One might see this as follows for degree $deg L=0$ (if $deg L\neq 0$ one should take a $f$-invariant metric of volumne 1 compatible with the Riemann surface structure and unitary connections with (constant) curvature $2\pi deg L vol$): every holomorphic structure $D$ gives rise to an unique unitary flat connection $\nabla$ such that $D=1/2(\nabla+i*\nabla).$ Then the anti-holomorphic structure on $\bar L$ is given by $1/2(\nabla-i*\nabla)$ and, the nitary flat connection corresponding to the holomorphic structure on $f^* L$ is the connection $f^* \nabla.$ This connection is gauge equivalent to $\nabla.$ Thus the holomorphic structures are isomorphic.