Evaluating line integral using The Stokes' Theorem
It should simply be,
$ \displaystyle \iint_S (\nabla \times \vec F) \cdot \hat n ~d\sigma = \iint_{x^2 + y^2 \leq 1} (\nabla \times \vec F) \cdot (\nabla f) ~ dA$
$ \displaystyle = \iint_{x^2 + y^2 \leq 1} (\hat i - \hat j + \hat k) \cdot (- \hat j + \hat k) ~dA = 2 \pi$
It is unnecessary to normalize the normal vector as $|\nabla f|$ cancels out. Note that $d\sigma = |\nabla f| ~dA$ where $d \sigma$ is the surface area element and $dA$ is the area element of the projection in xy-plane.