The scenario of a metal-insulator transition driven by the onset of antiferromagnetic order in NaOsO3 calls for a trustworthy derivation of the underlying effective spin Hamiltonian. To determine the latter we rely on ab initio electronic-structure calculations, linear spin-wave theory, and comparison to experimental data of the corresponding magnon spectrum. We arrive this way to Heisenberg couplings that are less than or similar to 45 to less than or similar to 63% smaller than values presently proposed in the literature and Dzyaloshinskii-Moriya interactions in the region of 15% of the Heisenberg exchange J. These couplings together with the symmetric anisotropic exchange interaction and single-ion magnetocrystalline anisotropy successfully reproduce the magnon dispersion obtained by resonant inelastic x-ray scattering measurements. In particular, the spin-wave gap fully agrees with the measured one. We find that the spin-wave gap is defined from a subtle interplay between the single-ion anisotropy, the Dzyaloshinskii-Moriya exchange, and the symmetric anisotropic exchange interactions. The results reported here underpin the local-moment description of NaOsO3, when it comes to analyzing the magnetic excitation spectra. Interestingly, this comes about from a microscopic theory that describes the electron system as Bloch states, adjusted to a mean-field solution to Hubbard-like interactions.