what is the universal cover for $S_g$?
Solution 1:
1) Any manifold admit a universal cover and the dimension is the same ; 2) The only two simply connected surfaces are the plane and the sphere.
If you look at your compact surface of genus $2$ as a complex manifold, then there is the famous Uniformization Theorem which states that there are only 3 different simply connected complex manifold of dimension 1 : the plane, the sphere and the half-plane.