In a recent paper by Lucas and Das Sarma [Phys. Rev. B 97. 115449 (2018)], a solvable model of collective modes in two-dimensional metals was considered in the hydrodynamic regime. In the current work, we generalize the hydrodynamic theory to three-dimensional (3D) metals for which the calculation of sound modes in a strongly coupled quantum Coulomb plasma can be made explicit. The specific theoretical question of interest is how the usual linearly dispersing hydrodynamic sound mode relates to the well-known gapped 3D plasmon collective mode in the presence of long-range Coulomb interaction. We show analytically that both the zero sound in the collisionless regime and the first sound in the hydrodynamic region become massive in three dimensions, acquiring a finite gap because of the long-range Coulomb interaction, while their damping rates become quadratic in momentum. We also discuss other types of long-range potential, where the dispersion of sound modes is modified accordingly. The general result is that the leading-order hydrodynamic sound mode is always given by the leading-order plasmon frequency in the presence of long-range Coulomb interaction, but the next-to-leading-order dispersion corrections differ in hydrodynamic and collisionless regimes.