Herein, an interference-aware predictive aerial-and-terrestrial communication problem is studied, where an unmanned aerial vehicle (UAV) delivers some data payload to a few nodes within a communication deadline. The first challenge is the possible interference to the ground base stations (BSs) and users possibly at unknown locations. This paper develops a radio-map-based approach to predict the channel to the receivers and the unintended nodes. Therefore, a predictive communication strategy can be optimized ahead of time to reduce the interference power and duration for the ground nodes. Such predictive optimization raises the second challenge of developing a low-complexity solution for a batch of transmission strategies over T time slots for N receivers before the flight. Mathematically, while the proposed interference-aware predictive communication problem is non-convex, it is converted into a relaxed convex problem, and solved by a novel dual-based algorithm, which is shown to achieve global optimality at asymptotically small slot duration. The proposed algorithm demonstrates orders of magnitude saving of the computational time for moderate T and N compared to several existing solvers. Simulations show that the radio-map-assisted scheme can prevent all unintended receivers with known positions from experiencing interference and significantly reduce the interference to the users at unknown locations.