A detailed theoretical investigation of structure, electronic and magnetic properties of a single-atom magnetic junction is presented. Our studies are based on the density functional theory and the multiple scattering approach. We concentrate on a Cu tip interacting with the Co adatom on Cu(001, and study this junction for the tip-substrate distances ranging from the tunneling to the contact regime. We demonstrate that both the atomic structure of the tip and the position of the adatom above the surface significantly depend on the tip-substrate distance. The adatom-surface distance is found to vary nonmonotonically as the tip approaches the surface. A pronounced change in the spin-polarized local density of states on the Co adatom, and a strong suppression of its magnetic moment are found when the tip is about 4 Å away from the surface.