This paper introduces a proactive Unmanned Aerial Vehicle (UAV) mobility management xApp for Open Radio Access Network (O-RAN) Near Real-Time Radio Intelligent Controller (Near-RT RIC) environments, employing Double Deep Q-Network (DDQN) reinforcement learning (RL) enhanced with transfer learning to optimise handover decisions for UAVs operating along predetermined flight trajectories. Unlike reactive approaches that respond to signal degradation, the proposed framework anticipates network conditions and minimises both outage probability and handover frequency through predictive optimisation. The system leverages centralised weight averaging to consolidate knowledge from multiple flight scenarios into a global model capable of generalising to previously unseen operational environments without extensive retraining. A comprehensive evaluation demonstrates that the proposed framework achieves a favourable trade-off between handover frequency and connectivity reliability, reducing handover events by up to 54.6% compared to greedy approaches while maintaining outage probability at practically negligible levels. The results validate the effectiveness of intelligent learning-based approaches for UAV mobility management in next-generation O-RAN architectures, thereby contributing to seamless integration of aerial user equipment into cellular networks.