  On the basis of the generalized eikonal approximation we study the exclusive
reactions $\bar p d \to J/\psi\, n$ and $\bar p d \to \psi^\prime\, n$ in
vicinity of the thresholds for charmonium production on a free proton target.
It is shown that the rescattering of the incoming antiproton and outgoing
charmonium on the spectator neutron leads to a depletion of the charmonium
production at low- and to an enhancement at high transverse momenta. This is in
qualitative agreement with previous studies of hard proton knockout in
proton-deuteron collisions. We analyze different physical sources of
uncertainty which may influence the extraction of the total charmonium-neutron
cross section. The color transparency effect for the incoming $\bar p$ largely
compensates the influence of charmonium rescattering both at low and high
transverse momenta. Different choices of the deuteron wave function lead to
significant uncertainties at high transverse momenta. As an outcome of the
calculations of charmonium production, we also provide predictions on the
production of open charm hadrons due to the dissociation of the charmonium on
the neutron. It is shown that the open charm production cross section is
proportional to the total charmonium-nucleon cross section and quite stable
with respect to the variation of other parameters of the model. We thus suggest
that open charm channels are most suited for future studies of
charmonium-nucleon interactions at PANDA with a deuteron target.
