Cascading failure modeling and analysis provide convenient tools for assessing and enhancing the robustness of power systems against severe power outages. In this paper, we apply a neural network-based classifier to predict the onset time of cascading failure. Onset time, which has been reported as the time when the number of component failure begins to rapidly increase in the failure propagation, serves as a crucial metric to evaluate the vulnerability of power systems to cascading failure. We formulate the prediction task as a multi-class classification problem and adopt a neural network-based classifier where topological and electrical information of a power system network can be exploited for learning. Experimental results on the UIUC 150-Bus power system demonstrate a high classification accuracy by only leveraging the initial states of power networks and the initial failure sets containing the power components to be tripped at the beginning of cascading failure.