By using the first-principles calculations within the density functional theory (DFT), we investigate the structure, antiferromagnetic-ferromagnetic phase transition and half-metallicity of half-Heusler compound Fe1-XCuxMnSb. For the ferromagnetic state, the calculated lattice constant follows the Vegard law, and total ferromagnetic moment agrees with Slater-Pauling rule as x<0.5 while suffers large deviation as x>0.5. As x=0.75, a novel magnetic phase transition occurs, which mainly derives from the competition between RKKY and superexchange magnetic coupling. Further analyses on density of states (DOS) reveal that the Fermi level can be adjusted by doped Cu concentration in ferromagnetic state, and d partial DOS of Mn atoms with different spin orientation compensate for each other that results in symmetric spin-up and spin-down band in antiferromagnetic state.