We use resonant inelastic x-ray scattering (RIXS) at the Cu L3 edge to measure the charge and spin excitations in the “half-stuffed” Cu-O planes of the cuprate antiferromagnet Ba2Cu3O4Cl2. The RIXS line shape reveals distinct contributions to the dd excitations from the two structurally inequivalent Cu sites, which have different out-of-plane coordinations. The low-energy response exhibits magnetic excitations. We find a spin-wave branch whose dispersion follows the symmetry of a CuO2 sublattice, similar to the case of the “fully stuffed” planes of tetragonal CuO (T-CuO). Its bandwidth is closer to that of a typical cuprate material, such as Sr2CuO2Cl2, than it is to that of T-CuO. We interpret this result as arising from the absence of the effective four-spin inter-sublattice interactions that act to reduce the bandwidth in T-CuO.