The Grad-type moment equations provide an extended hydrodynamic framework for describing the non-equilibrium behavior of rarefied polyatomic gases beyond the classical Navier-Stokes-Fourier equations. In this study, we present a numerical investigation of Grad-14 and Grad-17 moment equations to analyze their accuracy and applicability in capturing rarefaction effects, translational-rotational energy exchange, and non-equilibrium transport phenomena. The numerical scheme employs a high-order nodal discontinuous Galerkin method to solve the Grad-type equations for rarefied polyatomic flows. The results are compared with solutions from the direct simulation Monte Carlo (DSMC) method and experimental data to assess the predictive capability of the moment approach. Our analysis demonstrates that the Grad formulation offers superior accuracy over the Navier-Stokes-Fourier equations in capturing non-equilibrium phenomena at Mach numbers up to 1.6, particularly under strong velocity/temperature gradient conditions. Nevertheless, persistent non-physical sub-shock formations emerge as a critical limitation at Mach 2.0. Additionally, the study highlights the limitations of moment closure assumptions and their implications for extending higher-order moment theories to polyatomic gases.