The magnetism of CoPt nanostructures supported on the MgO(100) surface is investigated via first-principles simulations using 1D models. Nanostructures with L1(0) chemical ordering and cube-on-cube epitaxy are predicted to possess large magnetic moments and easy magnetization axis perpendicular to the surface. However, their magnetic anisotropy energy is roughly halved with respect to the bulk alloy due to a peculiar mixing of particle and support electronic states. The general factors at play in determining this behavior and the implications of these findings are discussed in view of designing room-temperature magnetic bits.