Driven by a magnetic field, the rotation of a particle near a wall can be rectified into a net translation. The particles thus actuated, or surface walkers, are a kind of active colloid that finds application in biology and microfluidics. Here, we investigate the motion of spherical surface walkers confined between two walls using simulations based on the immersed-boundary lattice Boltzmann method. The degree of confinement and the nature of the confining walls (slip vs no-slip) significantly affect a particle's translational speed and can even reverse its translational direction. When the rotational Reynolds number Re-omega, is larger than 1, inertia effects reduce the critical frequency of the magnetic field, beyond which the sphere can no longer follow the external rotating field. The reduction of the critical frequency is especially pronounced when the sphere is confined near a no-slip wall. As Re-omega, increases beyond 1, even when the sphere can still rotate in the synchronous regime, its translational Reynolds number Re-T no longer increases linearly with Re-omega, and even decreases when Re-omega exceeds similar to 10.