Inter-grain nanobridges of the $MgB_2$ superconductor have been fabricated by focused-ion-beam(FIB) and their electrical transport properties were studied. The $MgB_2$ film was prepatterned into microbridges by a standard argon ion milling technique and then FIB-patterned into 100 nm$ imes$100 nm bridges. Current-voltage characteristics showed a strong flux-flow type behavior at all temperatures with a trait of Josephson coupling near $T_c$. At low temperatures, the curves showed a two-step resistance-doubled transition with occasional hysteresis. The resistance-doubling transition is believed to be due to a two-channel flux-flow effect. The temperature-dependent critical current data showed $I_c(T){propto}(1-T/T_c)^2$ near $T_c$, same as a normal barrier junction, and $I_c(T){propto}(1-T/T_c)^{1.2}$ at low temperatures, similar to that of a film.