Neural State Classification (NSC) is a recently proposed method for runtime predictive monitoring of Hybrid Automata (HA) using deep neural networks (DNNs). NSC trains a DNN as an approximate {\em reachability predictor} that labels a given HA state x as positive if an unsafe state is reachable from x within a given time bound, and labels x as negative otherwise. NSC predictors have very high accuracy, yet are prone to prediction errors that can negatively impact reliability. To overcome this limitation, we present Neural Predictive Monitoring (NPM), a technique based on NSC and conformal prediction that complements the predictions with statistically sound estimates of uncertainty. This yields principled criteria for the rejection of predictions likely to be incorrect, without knowing the true reachability values. We also present an active learning method that significantly reduces both the NSC predictor's error rate and the percentage of rejected predictions. Our approach is highly efficient, with computation times on the order of milliseconds, and effective, managing in our experimental evaluation to successfully reject almost all incorrect predictions.