We consider convergence properties of the long-term behaviors with respect to the coefficient of the stochastic term for a nonautonomous stochastic $p$-Laplacian lattice equation with multiplicative noise. First, the upper semi-continuity of pullback random $(\ell^2,\ell^q)$-attractor is proved for each $q\in[1,+\infty)$. Then, a convergence result of the time-dependent invariant sample Borel probability measures is obtained in $\ell^2$. Next, we show that the invariant sample measures satisfy a stochastic Liouville type equation and a termwise convergence of the stochastic Liouville type equations is verified. Furthermore, each family of the invariant sample measures is turned out to be a sample statistical solution, which hence also fulfills a convergence consequence.