In this paper, transverse magnetic (TM) propagation modes of surface plasmon polaritons (SPPs) in graphene micro/nano ribbons are exhaustively characterized by accounting for the finite lateral dimensions of graphene, screening of Fermi level in multilayer graphene stack, and the impact of dielectric permittivity and the associated charge impurities at the dielectric-graphene interface. Fermi level screening leads to a non-uniform carrier density across multiple layers, which changes the electron relaxation rate and considerably alters the complex dynamical conductivity of multilayer GNRs. It is shown that ignoring the screening effects in multilayer GNRs overestimates both the SPP propagation length and its propagation velocity. Graphene plasmonic interconnects are envisaged as low energy, high frequency on-chip interconnects for future technology nodes. Simulations are performed over a broad frequency spectrum to identify the merits of future graphene plasmonic interconnects over the conventional electrical Cu/low-κ at a minimum feature size of 10 nm. Using energy-per-bit as a figure-of-merit, a range of SPP propagation lengths is identified for graphene plasmonic interconnects to outperform Cu interconnects.