We present a comprehensive first-principles investigation of a hypothetical cubic $Pm\overline{3}m$ phase of the ternary hydride NaAlH$_3$, focusing on its lattice dynamics, electronic structure, and electron-phonon-mediated superconducting properties at ambient pressure. Using density functional theory and the Migdal-Eliashberg formalism, we find an exceptionally strong electron-phonon coupling ($\lambda=2.23$), resulting in a superconducting critical temperature of up to $73.7$ K for a Coulomb pseudopotential $\mu^* = 0.1$. Phonon dispersion calculations, complemented by \textit{ab initio} molecular dynamics simulations, indicate dynamic and thermal stability within the adopted theoretical framework. The electronic structure exhibits a metallic character with substantial contributions from Al- and Na-derived states at the Fermi level. The resulting superconducting gap ratio ($2\Delta(0)/k_B T_c \approx 4.8$) and specific heat jump ($\Delta C/\gamma T_c \approx 2.2$) significantly exceed BCS weak-coupling predictions, highlighting the strong-coupling nature of superconductivity in this hypothetical phase.