Core-collapse supernovae provide natural laboratories for the production of new light particles. In particular, axion-like particles (ALPs) can be constrained via SN1987A cooling arguments. However, significant astrophysical and nuclear uncertainties imply that such bounds may vary strongly depending on modeling choices, even when expensive simulations are employed. In this context, semi-analytic methods offer a simple and fast alternative for deriving new-physics constraints. Building on a previous semi-analytic framework, in which proto-neutron star (PNS) observables are expressed in terms of six global PNS parameters, we include a finite ALP mass in the calculation and derive bounds in the axion-nucleon coupling versus mass plane. The obtained bounds are in good agreement with previous results from numerical simulations, demonstrating the robustness of the method. We also illustrate the sensitivity of the bounds to different PNS parameter calibrations, nuclear effects and cooling exclusion criteria.