By using first-principles calculations based on density-functional theory, the stability, magnetic and electronic properties of Fe and Co monoatomic chains encapsulated into copper nanotube are systematically investigated. The binding energies of the hybrid structures are remarkably higher than those of corresponding freestanding TM chains, indicating the TM chains are significantly stabilized after encapsulating into copper nanotube. The formed bonds between outer Cu and inner TM atoms show some degree of covalent bonding character. The magnetic ground states of Fe@CuNW and Co@CuNW hybrid structures are ferromagnetic, and both spin and orbital magnetic moments of inner TM atoms have been calculated. The magnetocrystalline anisotropy energies (MAE) of the hybrid structures are enhanced by nearly fourfold compared to those of corresponding freestanding TM chains, indicating that the hybrid structures can be used in ultrahigh density magnetic storage. Furthermore, the easy magnetization axis switches from that along the axis in freestanding Fe chain to that perpendicular to the axis in Fe@CuNT hybrid structure. The large spin polarization at the Fermi level also makes the hybrid systems interesting as good potential materials for spintronic devices.