The spin Hall effect is a paradigmatic manifestation of spin-orbit-coupled transverse transport, in which particles with different spin components acquire opposite transverse responses under a longitudinal drive. Ultracold atoms in optical lattices provide a clean and highly controllable platform for exploring such Hall-type dynamics, because the lattice geometry, artificial gauge field, spin-orbit coupling, and interaction strength can be engineered independently. In this work, we propose and numerically analyze a theoretical scheme for simulating spin-Hall-type dynamics with ultracold atoms in a quasi-one-dimensional optical lattice. The system is described by a ladder geometry, where two long-lived orbital states serve as a synthetic dimension. Rashba-type spin-orbit coupling is introduced together with a linear potential gradient along the physical lattice direction. Starting from the ground state without spin-orbit coupling and external driving, we quench on both the Rashba coupling and the linear potential, and calculate the subsequent non-equilibrium dynamics using the time-dependent density matrix renormalization group method. The dynamical response is characterized by the longitudinal particle current, the transverse spin current, and the transverse spin polarization. Our results show that the linear potential drives directional atomic transport along the longitudinal direction, while the Rashba coupling converts this motion into a spin-dependent transverse response. Consequently, opposite transverse currents emerge for the two spin components, giving rise to a finite transverse spin polarization. We further examine how the spin-Hall-type response depends on the Rashba coupling strength, the linear potential gradient, the system size, and the on-site interaction. The transverse spin polarization is enhanced by increasing the spin-orbit coupling and the longitudinal driving strength within the investigated parameter regime, whereas repulsive interactions tend to suppress the transverse response by modifying the local occupation and many-body correlations. These results provide a feasible theoretical route for simulating spin Hall physics in ultracold atomic systems and offer insight into non-equilibrium spin transport in engineered quantum matter.