The industry is pursuing Urban Air Mobility (UAM) to offer short-range, point-to-point air transportation services in metropolitan areas. This concept has the potential to mitigate ground-based congestion and improve transportation efficiency. To ensure the safety of UAM operations, the airborne navigation system must provide accurate and reliable navigation information. However, the design of such navigation systems remains not fully addressed in prior studies. Accordingly, this paper firstly presents a high-accuracy navigation system that employs Global Navigation Satellite Systems (GNSS)/ Inertial Navigation System (INS) tight integration and exploits differential GNSS corrections. Then, to protect against the potential measurement faults caused by complex signal reflections in urban environments, an Aircraft Autonomous Integrity Monitoring (AAIM) framework is developed by applying multiple hypotheses solution separation to the Kalman filter-based integration architecture. This framework offers two capabilities of essential importance to ensuring safe UAM operations: real-time fault detection and rigorous navigation integrity quantification. The proposed algorithms are validated by various simulations, and the results suggest promising performance.