We theoretically study the scattering of spin waves from a sharp domain wall (DW) in an antiferromagnetic spin chain. While the continuum model for an antiferromagnetic material yields the well-known result that spin waves can pass through a wide DW with no reflection, here we show that, based on the discrete spin Hamiltonian, spin waves are generally reflected by a DW with a reflection coefficient that increases as the DW width decreases. Remarkably, we find that, in the interesting case of an atomically sharp DW, the reflection of spin waves exhibits strong dependence on the state of circular polarization of the spin waves, leading to mainly reflection for one polarization while permitting partial transmission for the other, thus realizing an atomic-scale spin-wave polarizer. The polarization of the transmitted spin wave depends on the orientation of the spin in the sharp DW, which can be controlled by an external field or spin torque. Our utilization of a sharp antiferromagnetic DW as an atomic-scale spin-wave polarizer leads us to envision that ultrasmall magnetic solitons such as DWs and skyrmions may enable realizations of atomic-scale spin-wave scatterers with useful functionalities.