  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 $Br[P_{cs}(4459)^0 \to {J/\psi}\Lambda]$ 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.
