Unmanned aerial vehicle (UAV) assisted edge caching has been emerged as a promising solution to alleviate network congestion, which can provide users with their desired contents with reduced latency. For achieving effective UAV-assisted edge caching, how to jointly design the trajectory and caching strategy is critical, which, however, is not straightforward due to the heterogeneous and uncertain demand in the network. In this paper, aiming at maximizing the reduced delay brought by the UAV-assisted caching, we propose a proactive joint strategy on trajectory and caching for the UAV, where the demand uncertainty is particularly studied. Specifically, by regarding the demand on each content as a random variable, we formulate the strategy design as a risk-averse stochastic optimization problem to make the network performance guaranteed under certain confidence level. Different from most existing works assuming the perfect distributional information is available to deal with the uncertainty, we develop a data-driven approach based on the first and second order statistics to achieve a distributionally robust (DR) solution, which can make the strategy trustworthy with guaranteed network performance even though the specific distributional information is unknown. Simulation results have demonstrated the effectiveness of the proposed DR strategy.