  Forward rapidity $J/\psi$ meson production in proton-nucleus collisions can
be an important constraint of descriptions of the small-$x$ nuclear
wavefunction. In an earlier work we studied this process using a dipole cross
section satisfying the Balitsky-Kovchegov equation, fit to HERA inclusive data
and consistently extrapolated to the nuclear case using a standard Woods-Saxon
distribution. In this paper we present further calculations of these cross
sections, studying the mean transverse momentum of the meson and the dependence
on collision centrality. We also extend the calculation to backward rapidities
using nuclear parton distribution functions. We show that the parametrization
is overall rather consistent with the available experimental data. However,
there is a tendency towards a too strong centrality dependence. This can be
traced back to the rather small transverse area occupied by small-$x$ gluons in
the nucleon that is seen in the HERA data, compared to the total inelastic
nucleon-nucleon cross section.
