Temperature and pressure variation of Xe-129 chemical shift and longitudinal relaxation rate characterize the organization and dynamical properties of xenon gas confined within the porous network of Vycor. The chemical shift of confined xenon yields an adsorption enthalpy (5.3 +/- 0.5 kT/mol) in agreement with calorimetric measurements. Molecular dynamics simulation of gas diffusion is required to interpret the pathology of the spectral density, responsible for the large difference between the longitudinal and transverse relaxation rates. This work illustrates the high potentiality of Xe-129 NMR relaxation studies to probe the influence of pore morphology and topology on the diffusion of confined fluid.