Structural and magnetic properties of self-assembled Co nanorods on a Cu(110)-(2x3)N surface have been investigated by low-energy electron diffraction (LEED), Auger-electron spectroscopy (AES), magneto-optical Kerr effect (MOKE), and x-ray magnetic circular dichroism (XMCD). The LEED observation confirms that the Co nanorod grows epitaxially along the [1[1]0] axis and its interval exhibits the (1x6) periodicity. The AES clarifies that the N atom locates always at the surface even after 5 monolayer (ML) Co deposition. Angle dependent magnetization curves of the Co nanorods recorded by MOKE and XMCD show that the magnetic easy axis is perpendicular to the rod within the substrate plane, irrespective of the Co thickness down to 0.8 ML. This implies that the magnetic anisotropy is not dominated by the shape anisotropy but by the magnetocrystalline anisotropy. The XMCD sum-rule analysis reveals significant enhancement of the orbital magnetic moment along the easy axis compared to the hard axes. The magnetocrystalline anisotropy is found to be directly related to the anisotropy of the orbital magnetic moment.