This work attempts to present mathematical modeling of the dynamics of the hydroelastic system consisting of a hollow cylinder with inhomogeneous initial stresses and a compressible barotropic inviscid fluid which filled the interior of the cylinder. The motion of the cylinder is described with the 3D linearized theory of elastic waves in bodies with initial stresses; however, the flow of the fluid is described with the linearized Euler equations for compressible inviscid fluid. Formulations of the boundary conditions on the outer surface of the cylinder and compatibility conditions between the cylinder and fluid on the interior surface of the cylinder are presented. Concrete formulations are made for the axisymmetric case, and general aspects of the solution methods of the formulated problems are considered. Under these considerations, two types of problem are selected: (1) the axisymmetric longitudinal wave propagation and (2) fluid flow profile on the outer free surface of the cylinder—the “circular” radial moving load. Numerical results are presented and discussed for the first type of problem. In particular, it is established that the character of the influence of the inhomogeneous initial stresses on the wave propagation velocity in the hydroelastic system under consideration depends on the dimensionless wavenumber. The perspective of future investigations is also presented and discussed.