Charmonium production is extensively studied experimentally and theoretically since its discovery in 1974. It provides a testing ground for perturbative and non-perturbative aspects of QCD calculations in high-energy hadronic collisions. In heavy-ion collisions, it is used as a probe of the quark gluon plasma and in proton-nucleus collisions, it gives insight on the cold nuclear medium effects that modify its production w.r.t. its normalised production in proton-proton collisions. This work focuses on charmonium production in proton-nucleus collisions and initial or final state effects that can occur in such collisions are described in the first part. In the second part, the measurements performed with the ALICE experiment at the LHC in p-Pb collisions are presented and compared to theoretical calculations.They are as well discussed together with open and hidden heavy flavour measurements obtained at the LHC and, at lower energy, at RHIC. Finally, the third part describes the physics opportunities to study charmonium production in pA collisions in a fixed-target mode at theLHC. The general motivations for a fixed-target experiment at the LHC are presented, as well as the possible and current technical implementations. Expected performances in case of charmonium production in pA collisions are also discussed.