We study the J/Ψ production from nuclei near the kinematic threshold within the collision model, based on the nuclear spectral function, for incoherent primary proton-nucleon charmonium creation processes. The model takes into account the initial proton and final J/Ψ absorption, target nucleon binding and Fermi motion, the formation length of J/Ψ mesons as well as the effect of their nuclear mean-field potential on these processes. We calculate the A-dependences of the absolute and relative (transparency ratio) charmonium yields as well as its excitation function within the different scenarios for the J/ΨN absorption cross-section, for the J/Ψ formation length and for J/Ψ in-medium modification. We demonstrate that the former observables, on the one hand, are not practically affected by the charmonium formation length effects and, on the other hand, they are appreciably sensitive to the genuine J/ΨN absorption cross-section at beam energies of interest, which means that these observables can be useful to help determine the above cross-section from the comparison of the results of our calculations with the future data from the CBM experiment at FAIR-the upcoming accelerator facility at GSI-Darmstadt, Germany. We also show that the excitation function for J/Ψ subthreshold production in pA reactions, reveals some sensitivity to adopt its in-medium modification scenarios. Therefore, such observable may be an important tool to get valuable information on the charmonium in-medium properties. [ABSTRACT FROM AUTHOR]