The EPR experiment measures the interaction of unpaired spin density ($\rho(\alpha)$) at the nuclei. This is quantified by an equation called the Fermi contact interaction.
If you have some prior knowledge of "where to put the dot" in your structure (highly localized), then generally you can assume that the EPR pattern is split most by that nucleus. Splitting patterns are determined by standard rules in accord with the Zeeman effect.
The nucleus must have a non-zero nuclear spin to be observable. Certain nuclei such as hydrogen-1 and many transition metals have spin. It is very common that isotopic impurities (e.g. carbon-13) appear though the most prevalent isotope is EPR-silent.
(There are other complications, such as nuclear quadrupole moment and broadening due to exchange of chemical sites.)
An additional issue is that the spin density is generally spread over the molecule, and interacts with multiple nuclei. It turns out to be easier to measure the spectrum, and then rationalize it given the particular structure and nuclear composition.
Accurate prediction of EPR spectra is a rather difficult problem, as quantum chemical calculations generally don't emphasize basis function placement at the nucleus, as they tend to focus on the "valence" and not the cusp-like nature of the radial distribution. (Here, Slater-type orbitals would be better in principle than Gaussian-type, but STOs of sufficient number and of high-enough quality of Hamiltonian are virtually impossible). Finally, solvent and powder effects can render the gas-phase ab initio calculations virtually irrelevant.
EPR simulation/interpretation of fine structure is much more difficult than NMR spectra, because in NMR spectra, one has the chemical shift to conveniently move splitting patterns out of the way. In EPR, the splittings are often "piled" on top of each other, and make it necessary to use computer simulation to obtain precise fits.
In your specific example, I would identify the nuclei with spin (probably Cu, certainly phosphorus-31, and then hydrogen-1). Then, identify where you think the unpaired density will be, and then take the nuclear spin + 1 and that will be the "guesstimated" isotropic spectrum. The pattern will follow some variant Pascal's triangle if I=1/2, otherwise, you have to figure it out by writing out the number of possible configurations.
Edit: I have not even started on the characteristics of triplet spectra, as here the electrons interact. One thing you can hope for is that all of your electrons are paired (singlet), and so it's EPR silent.