In this paper, we investigate the stability of pullback random attractors for non-autonomous stochastic $p$-Laplacian lattice systems, which are influenced by random viscosity and multiplicative white noise. Under appropriate conditions, we first prove the existence and uniqueness of these pullback random attractors and then establish their backward compactness. To ensure their measurability, we demonstrate the equivalence of two different classes of attractors across two distinct universes. Finally, we examine the asymptotic stability of these pullback random attractors by assuming that the time-dependent external forcing term converges to a time-independent external force as time approaches negative infinity.