A systematic study of the binding and adsorption energies of the surface atoms on different sites of Cun clusters with tetrahedral shape is performed using density functional theory. The atom number n ranging from 10 to 120 and atop adsorption of carbon monoxide on these clusters are tested. The obvious size and site dependence of the binding energy Ec-x of the surface atoms and their adsorption energy Ead-x for CO molecule respectively located on vertex (v), edge (e), and (111) facet (f) are observed, with x meaning site and being v, e and f. Even though the obvious size dependence on both Ec-x and Ead-x exist, their regular changes with size disappear. For the site influence on the surface atom in these Cu clusters, the binding energies and adsorption energies have the sequence: Ec-v > Ec-e > Ec-f and Ead-v > Ead-e > Ead-f, but Cu35 and Cu56 are the exceptions in Ead-x. In these clusters, Cu20 shows the lowest Ec-v, Ec-e and Ec-f values, while the strongest adsorption happens on the edge of Cu35 and the weakest one on Cu84 facet site. This means a scaling relation between Ec-x and Ead-x is broken in smaller systems, and both size and site are required knowing in understanding the atomic properties on small clusters.