  We present first-order post-Newtonian (1PN) approximations of a general
imperfect fluid and of an axion as a coherently oscillating massive scalar
field, both in the cosmological context. For the axion, using the Klein
transformation and Madelung transformation we derive the Schr\"odinger and
Madelung hydrodynamic formulations, respectively, in exact covariant way and to
1PN order. Complete sets of equations for the 1PN formulations are derived
without fixing the temporal gauge condition. We study the linear instability in
cosmology and a static limit for both fluid and axion; these are presented
independently of the gauge condition to 1PN order, thus are naturally
gauge-invariant.
