A discretized version of the polymer RISM (PRISM) formalism using atomic closures has been applied to study the structural and thermal properties (the equation of state and liquid-gas spinodal) of the polymeric nearest-neighbor cubic lattice fluid. The obtained structural and equation of state properties are compared to NpT Monte Carlo simulations. From the comparison to the simulations, it follows that the predictions of the theory for both the structural correlations and the equation of state of the lattice fluid gradually worsen with increasing interaction strength. We find that the equation of state compares unfavorably with a, more conventional, nonrandom mixing-type theory based on the quasi-chemical approximation. We believe that the deterioration of the theory with increasing interaction strength is inherent in the PRISM plus atomic closure formalism and not typical for the lattice model studied in this work.