Technological advancements in unmanned aerial vehicles (UAVs) have revolutionized various industries, enabling the widespread adoption of UAV-based solutions. In engineering management, UAV-based inspection has emerged as a highly efficient method for identifying hidden risks in high-risk construction environments, surpassing traditional inspection techniques. Building on this foundation, this paper delves into the optimization of UAV inspection routing and scheduling, addressing the complexity introduced by factors such as no-fly zones, monitoring-interval time windows, and multiple monitoring rounds. To tackle this challenging problem, we propose a mixed-integer linear programming (MILP) model that optimizes inspection task assignments, monitoring sequence schedules, and charging decisions. The comprehensive consideration of these factors differentiates our problem from conventional vehicle routing problem (VRP), leading to a mathematically intractable model for commercial solvers in the case of large-scale instances. To overcome this limitation, we design a tailored variable neighborhood search (VNS) metaheuristic, customizing the algorithm to efficiently solve our model. Extensive numerical experiments are conducted to validate the efficacy of our proposed algorithm, demonstrating its scalability for both large-scale and real-scale instances. Sensitivity experiments and a case study based on an actual engineering project are also conducted, providing valuable insights for engineering managers to enhance inspection work efficiency.