Nucl.Phys.A 1023(2022)122452 We study the near-threshold $J/\psi$ meson production on protons and nuclei by considering incoherent direct non-resonant $K^-p \to {J/\psi}\Lambda$ and two-step resonant ${K^-}p \to P_{cs}(4459)^0 \to {J/\psi}\Lambda$ charmonium production processes. We calculate the absolute excitation functions, energy and momentum distributions for the non-resonant, resonant and for the combined (non-resonant plus resonant) production of $J/\psi$ mesons off protons as well as off carbon and tungsten target nuclei at near-threshold incident antikaon energies by considering these elementary production channels as well as by assuming the spin-parity assignment of the hidden-charm resonance $P_{cs}(4459)^0$ with strangeness as $J^P=(3/2)^-$ within six different scenarios for the branching ratio $BrP_{cs}(4459)^0 \to {J/\psi}\Lambda$ of the decay $P_{cs}(4459)^0 \to {J/\psi}\Lambda$. We show that the combined observables considered reveal definite sensitivity to these scenarios, which means that they may be an important tool to provide further evidence for the existence of the pentaquark $P_{cs}(4459)^0$ resonance and to get valuable information on its decay rate to the ${J/\psi}\Lambda$ final state. Their measurements could be performed in the future at the J-PARC Hadron Experimental Facility.