The SuperFGD detector will be a novel and important upgrade to the ND280 near detector for both the T2K and Hyper-Kamiokande projects. The main goal of the ND280 upgrade is to reduce systematic uncertainties associated with neutrino flux and cross-section modeling for future studies of neutrino oscillations using the T2K and Hyper-Kamiokande experiments. The upgraded ND280 detector will be able to perform a full exclusive reconstruction of the final state from neutrino-nucleus interactions, including measurements of low momentum protons, pions and for the first time, event-by event measurements of neutron kinematics. Precisely understanding the time resolution is critical for the neutron energy measurements and hence an important factor in reducing the systematic uncertainties. In this paper we present the results of time resolution measurements made with the SuperFGD prototype that consists of 9216 plastic scintillator cubes (cube size is 1 cm$^{3}$) readout with 1728 wavelength-shifting (WLS) fibers along the three orthogonal directions. We used data from a muon beam exposure at CERN. A time resolution of 0.97 ns was obtained for one readout channel after implementing the time calibration with a correction for time-walk effects. The time resolution improves with increasing energy deposited in a scintillator cube, improving to 0.87 ns for large pulses. Averaging two readout channels for one scintillator cube further improves the time resolution to 0.68 ns implying that signals in different channels are not synchronous. In addition the contribution from the time sampling interval of 2.5 ns is averaged as well. Most importantly, averaging time values from N channels improves the time resolution by ∼ 1/√(N). For example, averaging the time from 2 scintillator cubes with 2 fibers each improves the time resolution to 0.47 ns which is much better than the intrinsic electronics time resolution of 0.72 ns in one channel due to the 2.5 ns sampling window. This indicates that a very good time resolution should be achievable for neutrons since neutron recoils typically interact with several scintillator cubes and in addition produce larger signal amplitudes than muons. Measurements performed with a laser and a wide-bandwidth oscilloscope in which the contribution from the electronics time sampling window was removed demonstrated that the time resolution obtained with the muon beam is not far from the theoretical limit. The intrinsic time resolution of a scintillator cube and one WLS fiber is about 0.67 ns for signals of 56 photo electrons which is typical for minimum ionizing particles.