In unmanned aerial vehicle (UAV) applications, the UAV's limited energy supply and storage have triggered the development of intelligent energy-conserving scheduling solutions. In this paper, we investigate energy minimization for UAV-aided communication networks by jointly optimizing data-transmission scheduling and UAV hovering time. The formulated problem is combinatorial and non-convex with bilinear constraints. To tackle the problem, firstly, we provide an optimal algorithm (OPT) and a golden section search heuristic algorithm (GSS-HEU). Both solutions are served as offline performance benchmarks which might not be suitable for online operations. Towards this end, from a deep reinforcement learning (DRL) perspective, we propose an actor-critic-based deep stochastic online scheduling (AC-DSOS) algorithm and develop a set of approaches to confine the action space. Compared to conventional RL/DRL, the novelty of AC-DSOS lies in handling two major issues, i.e., exponentially-increased action space and infeasible actions. Numerical results show that AC-DSOS is able to provide feasible solutions, and save around 25-30% energy compared to two conventional deep AC-DRL algorithms. Compared to the developed GSS-HEU, AC-DSOS consumes around 10% higher energy but reduces the computational time from second-level to millisecond-level.