The structure of a 2D grid-like copper(II) complex [Cu$(NCO)_2$(pyz)](pyz=pyrazine) (1) consists of 1D chains of Cu-pyz units connected by double end-on (EO) cyanato bridges. Each Cu(II) ion has a distorted octahedral coordination, completed by the four EO cyanato and two pyrazine ligands. Magnetic interactions through EO cyanato and pyrazine bridges in 1 are discussed on the basis of DFT broken-symmetry calculations at the B3LYP level. For model dicopper(II) complexes I (bridged by cyanato) and II (bridged by pyrazine), electronic structure calculations reproduce very well the experimental couplings for the S = 1/2 ferromagnetic and antiferromagnetic exchange-coupled 2D system: the calculated exchange parameters J are +1.25 $cm^{-1}$ and -3.07 $cm^{-1}$ for I and II, respectively. The $sigma$ orbital interactions between the Cu $x^2-y^2$ magnetic orbitals and the nitrogen lone-pair orbitals of pyrazine are analyzed from the viewpoint of through-bond interaction. The energy splitting of 0.106 eV between two SOMOs indicates that the superexchange interaction should be antiferromagnetic in II. On the other hand, there are no bridging orbitals that efficiently connect the two copper(II) magnetic orbitals in I because the HOMOs of the basal-apical NCO bridge do not play a role in the formation of overlap interaction pathway. The energy separation in the pair of SOMOs of I is calculated to be very small (0.054 eV). This result is consistent with the occurrence of weakly ferromagnetic properties in I.