You are right, I went to fast. The Milnor fiber is the fiber of $U_x\setminus (C \cap U_x)$ to a punctured disc. Now an argument using the Serre spectral sequence should give you $H^1(U_x \setminus (C\cap U_x))$ has rank one if and only if $H^1$ of the Milnor fiber is trivial. (There are also exact sequences which for fibrations relate the fundamental group of the total space, the fiber and the base. See e.g., page 73 of Dimca's book Singularities and Topology of Hypersurfaces. Hence you may be able to avoid spectral sequences)

To prove the lemma: Since $C$ is a curve in a smooth surface and you have a local statement you can reduce to the case of plane curves. For plane curves you can prove the lemma by noting that the first betti number of the Milnor fiber equals the Milnor number of the singularity and that $H^1$ of the Milnor fiber is isomorphic with the abelianization of the local fundamental group. Moreover the Milnor number is zero if and only if $C$ is smooth at $x$.