Black holes in anti-de Sitter spacetime provide an important testing ground for both gravitational and field-theoretic phenomena. In particular, the study of perturbations can be useful to further our understanding regarding certain physical processes, such as superradiance, or the dynamics of strongly coupled conformal field theories through the holographic principle. In this work we continue our systematic study of the ultraviolet instabilities of black-hole quasinormal modes, built on the characterization of the latter as eigenvalues of a (spectrally unstable) nonselfadjoint operator and using the pseudospectrum as a main analysis tool, extending our previous studies in the asymptotically flat setting to anti-de Sitter asymptotics. Very importantly, this step provides a singularly well-suited probe into some of the key structural aspects of the pseudospectrum. This is a consequence of the specific features of the Schwarzschild-anti-de Sitter geometry, together with the existence of a sound characterization by Warnick of quasinormal modes as eigenvalues, that is still absent in asymptotic flatness. This work focuses on such structural aspects, with an emphasis on the convergence issues of the pseudospectrum and, in particular, the comparison between the hyperboloidal and null slicing cases. As a physical by-product of this structural analysis we assess, in particular, the spectral stability of purely imaginary "hydrodynamic" modes, which appear for axial gravitational perturbations, that become dominant when the black-hole horizon is larger than the anti-de Sitter radius. We find that their spectral stability, under perturbations, depends on how close they are to the real axis, or conversely how distant they are from the first oscillatory overtone.