Landing a multi-rotor uncrewed aerial vehicle (UAV) on a moving target in the presence of partial observability, due to factors such as sensor failure or noise, represents an outstanding challenge that requires integrative techniques in robotics and machine learning. In this paper, we propose embedding a long short-term memory (LSTM) network into a variation of proximal policy optimization (PPO) architecture, termed robust policy optimization (RPO), to address this issue. The proposed algorithm is a deep reinforcement learning approach that utilizes recurrent neural networks (RNNs) as a memory component. Leveraging the end-to-end learning capability of deep reinforcement learning, the RPO-LSTM algorithm learns the optimal control policy without the need for feature engineering. Through a series of simulation-based studies, we demonstrate the superior effectiveness and practicality of our approach compared to the state-of-the-art proximal policy optimization (PPO) and the classical control method Lee-EKF, particularly in scenarios with partial observability. The empirical results reveal that RPO-LSTM significantly outperforms competing reinforcement learning algorithms, achieving up to 74% more successful landings than Lee-EKF and 50% more than PPO in flicker scenarios, maintaining robust performance in noisy environments and in the most challenging conditions that combine flicker and noise. These findings underscore the potential of RPO-LSTM in solving the problem of UAV landing on moving targets amid various degrees of sensor impairment and environmental interference.