  Recent observations indicate that magnetars may commonly reside in merging
compact binaries and at least part of fast radio bursts (FRBs) are sourced by
magnetar activities. It is natural to speculate that a class of merging neutron
star binaries may have FRB emitters. In this work, we study the observational
aspects of these binaries - particularly those with FRB repeaters, which are
promising multi-band and multi-messenger observation targets of radio
telescopes and ground based gravitational wave detectors as the former
telescopes can probe the systems at a much earlier stage in the inspiral than
the latter. We show that observations of FRB repeaters in compact binaries have
a significant advantage in pinning down the binary spin dynamics, constraining
neutron star equation of state, probing FRB production mechanisms, and testing
beyond standard physics. As a proof of principle, we investigate several mock
observations of FRB pulses originating from pre-merger neutron star binaries,
and we find that using the information of FRB arriving times alone, the
intrinsic parameters of this system (including the stellar masses, spins, and
quadrupole moments) can be measured with high precision, and the angular
dependence of the FRB emission pattern can also be well reconstructed. The
measurement of stellar masses (with an error of $\mathcal{O}(10^{-6}-10^{-5})$)
and quadrupole moments (with an error of $\mathcal{O}(1\%-10\%)$) may be an
unprecedented discriminator of nuclear equations of state in neutron stars. In
addition, we find the multi-band and multi-messenger observations of this
binary will be sensitive to alternative theories of gravity and beyond standard
models, e.g., dynamical Chern-Simons gravity and axion field that is coupled to
matter.
