Critical properties of the compact tree-dimensional $U(1)$ lattice gauge theory are explored at finite temperature on an asymmetric lattice. For vanishing value of the spatial gauge coupling one obtains an effective two-dimensional spin model which describes the interection between Polyakov loops. We study numerically the effective spin model for $N_t=1,4,8$ on lattices with spatial extension renging from $L=64$ to $L=256$. Our results indicate that the finite-temperature $U(1)$ lattice gauge theory belongs to the universality class of the two-dimensional $XY$ model, thus supporting the Svetitsky-Yaffe conjecture.