This article proposes an innovative Online Learning (OL) algorithm designed for efficient microgrid energy management, integrating Recurrent Neural Networks (RNNs), and Model Predictive Control (MPC) in an End-to-End (E2E) learning-based control architecture. The algorithm leverages the RNN capabilities to predict uncertain and possibly evolving profiles of electricity price, load demand, and renewable generation. These are then exploited in an integrated MPC optimization problem to minimize the overall microgrid electricity consumption cost while guaranteeing operation constraints. The proposed methodology incorporates a specifically designed online version of the Stochastic Weight Averaging (O-SWA) and Experience Replay (ER) methods to enhance OL capabilities, ensuring more robust and adaptive learning in real-time scenarios. In addition, to address the challenge of model uncertainty, a task-based loss approach is proposed by integrating the MPC optimization as a differentiable optimization layer within the Neural Network (NN), allowing the OL architecture to jointly optimize prediction and control performance. The performance of the proposed methodology is evaluated through extensive simulation results, showcasing its Transfer Learning (TL) capabilities across different microgrid sites, which are crucial for deployment in real microgrids. We finally show that our OL algorithm can be used to estimate the prediction uncertainty of the unknown profiles.