The tetragonally distorted antiperovskite Mn3CuN exhibits large magnetostriction of up to 2000 ppm, due to the ferromagnetic shape memory effect. Magnetocrystalline anisotropy, which is deeply involved in the magnetostriction, is investigated for Mn3CuN via quantitative evaluation of spin and orbital magnetic moments using soft x ray magnetic circular dichroism. Although Cu and Mn have finite magnetic contributions, the Cu contribution may be too small to dominate the magnetism of Mn3CuN. The orbital magnetic moment of Mn is found to be negligibly small, suggesting small magnetocrystalline anisotropy. Even with small magnetocrystalline anisotropy, the system exhibits ferromagnetic shape memory effects, presumably because the simultaneously small crystallographic anisotropy enhances the magnetic shear stress acting across the twinning plane.