The minimal $U(1)_X$ extension of the Standard Model (SM) is a well-motivated new physics scenario, where the anomaly cancellation requirement dictates the new neutral gauge boson ($Z^\prime$) couplings with the SM fermions in terms of two scalar charges ($x_H$ and $x_\Phi$). In this paper, we investigate the SM charged fermion pair production mechanism for different values of these scalar charges in the $U(1)_X$ scenario at future electron-positron colliders, i.e. $e^+e^-\to f\bar{f}$. Apart from the standard photon and $Z$ boson exchange for this process, this model features a $s$-channel (or both $s$ and $t$-channel for $f=e^-$) $Z^\prime$-boson exchange, which interferes with the SM processes. Considering the dilepton and dijet signatures from the heavy resonance we estimate the bounds on the U$(1)_X$ coupling $(g^\prime)$ and the $Z^\prime$ mass $(M_{Z^\prime})$. Considering the LEP-II results and prospective International Linear Collider (ILC) bounds on the effective scale for the four fermion interaction we estimate the reach on $M_{Z^\prime}/g^\prime$ for different center of mass energies. 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 which can show substantial deviations from the SM results, even for a multi-TeV $Z'$. This provides a powerful complementary way to probe the heavy $Z'$ parameter space beyond the direct reach of the Large Hadron Collider (LHC), as well as an effective way to determine the $U(1)_X$ charges.