If you have computed , then the surface integral on, for instance, the face of the cube with is: , where is a unit vector in direction (hence orthogonal to the face), meaning that you integrate the -component of over the face. And you have to do this for each of the 5 faces. Good luck!
Or you remember the title "using Stokes' thm to evaluate..." and write your surface integral (over : the cube minus the bottom face) as the integral over the boundary of , which is a square. This reduces the problem to the computation of four line integrals (and you don't need the curl). For instance, one of them is:
You also have to care about the orientation of the boundary: it is defined by the orientation of the surface.