In deformable shells conveying pulsatile flow, oscillatory pressure changes cause local movements of the fluid and shell wall, which propagate downstream in the form of a wave. In biomechanics, it is the propagation of the pulse that determines the pressure gradient during the flow at every location of the arterial tree. In this study, a woven Dacron vascular prosthesis is modelled as a transversely isotropic circular cylindrical shell described by means of nonlinear Novozhilov shell theory. Flexible boundary conditions are considered to simulate connection with the remaining tissue. Nonlinear vibrations of the shell conveying pulsatile flow and subjected to pulsatile pressure are investigated taking into account the effects of the pulse-wave propagation. An input oscillatory pressure at the shell entrance is considered and it propagates down the shell causing a wave motion within the shell where, as a consequence, the pressure gradient and the flow velocity are functions of both the axial coordinate and time. For the first time in literature, coupled fluid-structure Lagrange equations for a non-material volume with wave propagation in case of pulsatile flow are developed. The fluid is modeled as a Newtonian inviscid pulsatile flow and it is formulated using a hybrid model based on the linear potential flow theory and considering the unsteady viscous effects obtained from the unsteady time-averaged Navier-Stokes equations. Contributions of pressure and velocity changes' propagation are also considered in the pressure drop along the shell and in the pulsatile frictional traction on the internal wall in the axial direction. A numerical bifurcation analysis employs a refined reduced order model to investigate the dynamic behavior of a pressurized Dacron vascular graft conveying blood flow. A pulsatile time-dependent blood flow model is considered in order to study the effect of pressurization by applying the first and second harmonic of the physiological waveforms of velocity and pressure during the heart beating period. Geometrically nonlinear vibration response to pulsatile flow and transmural pulsatile pressure considering the propagation of pressure and velocity changes inside the shell are here presented via frequency-response curves and time histories. It is shown how traveling waves of pressure and velocity cause a delay in the radial displacement of the shell at different values of the axial coordinate. This study provides a deep insight into the currently unknown nonlinear behavior of vascular prostheses whose dynamic response can cause unwanted hemodynamic effects leading to failure. Indeed, it is well known that vascular prostheses mechanical properties are very different from those of natural arteries. In particular, the compliance mismatch between the host artery and the prosthesis causes a different wave speed resulting in a change in the performance of the cardiovascular system. In the near future, a more refined model to the one here presented will be applied to reproduce and compare the dynamic behavior of vascular prostheses and the human aorta, helping in vascular prostheses design and implementation.