  We investigate the extent to which resonances between an oscillating
background of ultra-light axion and a binary Keplerian system can affect the
motion of the latter. These resonances lead to perturbations in the
instantaneous time-of-arrivals, and to secular variations in the period of the
binary. While the secular changes at exact resonance have recently been
explored, the instantaneous effects have been overlooked. In this paper, we
examine the latter using N-body simulations including the external oscillatory
forcing induced by the axion background. While the secular effects are
restricted to a narrow width near the resonance, the instantaneous changes,
albeit strongest close to resonances, are apparent for wide range of
configurations. We compute the signal-to-noise ratio (SNR) as a function of
semi-major axis for a detection of axion oscillations through the R\{o} mer
delay. The latter can be extracted from the time-of-arrivals of binary pulsars.
The SNR broadly increases with increasing binary eccentricity in agreement with
the secular expectation. However, we find that it differs significantly from
the scaling a^{5/2} around the lowest orders of resonance. Future observations
could probe these effects away from resonances and, therefore, constrain a much
broader range of axion masses provided that binary pulsar systems are found
near the central region of our Galaxy, and that the time-or-arrival measurement
accuracy reaches < 10 ns
