 Solution

For a degree-one map, the sum of local degrees in every fiber is one. Therefore each fiber contains exactly one point and its local degree is one. There is no ramification, and the Riemann-Hurwitz formula gives $2g-2=-2$, so $g=0$. The map is a continuous bijection from compact $C$ onto the Hausdorff sphere and hence a homeomorphism. In fact its local degree-one coordinate inverses are holomorphic, so it is biholomorphic as well.