We investigate the effect of spin-orbit coupling on the in-gap bound states localized at magnetic impurities in multiband superconductors with unconventional (sign-changed) and conventional (sign-unchanged) s-wave pairing symmetry, which may be relevant to iron-based superconductors. Without spin-orbit coupling, for spin-singlet superconductors it is known that such bound states cross zero energy at a critical value of the impurity scattering strength and acquire a finite spin polarization. Moreover, the degenerate, spin-polarized, zero-energy bound states are unstable to applied Zeeman fields as well as a deviation of the impurity scattering strength away from criticality. Using a T-matrix formalism as well as analytical arguments, we show that, in the presence of spin-orbit coupling, the zero-energy bound states localized at magnetic impurities in unconventional, sign-changed, s-wave superconductors acquire surprising robustness to applied Zeeman fields and variation in the impurity scattering strength, an effect which is absent in the conventional, sign-unchanged, s-wave superconductors. Given that the iron-based multiband superconductors may possess a substantial spin-orbit coupling as seen in recent experiments, our results may provide one possible explanation to the recent observation of surprisingly robust zero bias scanning tunneling microscope peaks localized at magnetic impurities in iron-based superconductors provided the order parameter symmetry is sign changing s(+-)-wave.