In this study, we have investigated the effects of Mn dopant on the bulk trap levels and grain boundary characteristics of $Bi_2O_3$-based ZnO (ZB) varistor using admittance spectroscopy and dielectric functions (such as $Z^*,;Y^*,;M^*,;varepsilon^*$, and $tandelta$). Admittance spectra and dielectric functions show two bulk traps of $Zn_i^{..}$ (0.20 eV) and $V^{ullet}_o$ (0.29~0.33 eV) in ZnO-$Bi_2O_3-Mn_3O_4$ (ZBM). The barrier of grain boundaries in ZBM could be electrochemically single type. However, its thermal stability was slightly disturbed by ambient oxygen because the apparent activation energy of grain boundaries was changed from 0.79 eV at lower temperature to 1.08 eV at higher temperature. The grain boundary capacitance $C_{gb}$ was decreased slightly with temperature as 1.3~1.8 nF but resistance $R_{gb}$ decreased exponentially. The relaxation time distribution can result from the heterogeneity of the barriers constituting the varistor. It is revealed that Mn dopant in ZB reduced the heterogeneity of the barrier in grain boundaries and stabilized the barrier against the ambient temperature.