This study proposes a novel sandwich panel with an improved star-shaped honeycomb (SP-ISH) by replacing sharp corners with cubic struts and lengthening inclined ribs of traditional honeycomb. To more effectively analyze its special dynamic characteristics, the bending Poisson’s ratio was derived and effective engineering constants were obtained through asymptotic analysis of the energy functional stored in periodic unit cells. Based on this analysis, a two-dimensional equivalent downscaling model (2D-EDM) was developed to investigate the free vibration of SP-ISH under various boundary conditions, and the influence of initial deformation on free vibration was also discussed. The frequency-domain and time-domain analyses were employed to identify the resonance characteristics of the SP-ISH under harmonic loading. Our findings demonstrate that 2D-EDM can accurately predict the free and forced vibration of SP-ISH and improve computational efficiency by reducing DOFs. The results indicate that the maximum resonance response was obtained by applying excitation at the center of the sandwich panel. Additionally, under cyclic loading, a sharp change in local stress amplitude was observed in the facesheet-core interface region. Compared with traditional star-shaped honeycomb, the incorporation of extended ribs and added struts could significantly reduce the maximum resonance amplitude by 3.8%. Notably, the natural frequencies of SP-ISH were affected by initial compression (up to a 40.6% reduction) and tensile deformation (up to a 26.4% increase).