A spurt of progress in wireless power transfer (WPT) and mobile edge computing (MEC) provides a promising approach for Industrial Internet of Things (IIoT) to enhance the quality and productivity of manufacturing. Scheduling in such a scenario is challenging due to congested wireless channels, time-dependent energy constraints, complicated device heterogeneity, and prohibitive signaling overheads. In this article, we first propose an online algorithm, called energy-aware resource scheduling (ERS), to maximize the system utility comprising throughput and fairness, with consideration on both system sustainability and stability. Based on Lyapunov optimization and convex optimization techniques, the proposed algorithm achieves asymptotic optimality for heterogeneous IIoT systems without prior knowledge of network state information (NSI). Subsequently, we extend the ERS algorithm to a more realistic scenario where the overhead and delay of NSI feedbacks are non-negligible. The optimal scheduling decisions of the scenario are provided, and the optimality loss on system utility under outdated NSI is analyzed. The simulations verify our theoretical claims and demonstrate the gains of our proposed ERS algorithm over alternative benchmark schemes.