  The minimal $U(1)_X$ extension of the Standard Model (SM) is a well-motivated
new physics scenario, where anomaly cancellation dictates new neutral gauge
boson ($Z^\prime$) couplings with the SM fermions in terms of the $U(1)_X$
charges of the new scalar fields. We investigate the SM charged fermion
pair-production process for different values of these $U(1)_X$ charges at
future $e^-e^+$ colliders: $e^+e^-\to f\bar f$. Apart from the standard
$\gamma$ and $Z$-mediated processes, this model features additional $s$-channel
(or both $s$ and $t$-channel when $f=e^-$) $Z^\prime$ exchange which interferes
with the SM processes. We first estimate the bounds on the $U(1)_X$ coupling
$(g^\prime)$ and the $Z^\prime$ mass $(M_{Z^\prime})$ considering the latest
dilepton and dijet constraints from the heavy resonance searches at the LHC.
Then using the allowed values of $g^\prime$, we study the angular
distributions, forward-backward $(\mathcal{A}_{\rm{FB}})$, left-right
$(\mathcal{A}_{\rm{LR}})$ and left-right forward-backward
$(\mathcal{A}_{\rm{LR, FB}})$ asymmetries of the $f\bar{f}$ final states. We
find that these observables can show substantial deviations from the SM results
in the $U(1)_X$ model, thus providing a powerful probe of the multi-TeV
$Z^\prime$ bosons at future $e^+e^-$ colliders.
