Aggregating distributed energy resources in power systems significantly increases uncertainties, in particular caused by the fluctuation of renewable energy generation. This issue has driven the necessity of widely exploiting advanced predictive control techniques under uncertainty to ensure long-term economics and decarbonization. In this paper, we propose a real-time uncertainty-aware energy dispatch framework, which is composed of two key elements: (i) A hybrid forecast-and-optimize sequential task, integrating deep learning-based forecasting and stochastic optimization, where these two stages are connected by the uncertainty estimation at multiple temporal resolutions; (ii) An efficient online data augmentation scheme, jointly involving model pre-training and online fine-tuning stages. In this way, the proposed framework is capable to rapidly adapt to the real-time data distribution, as well as to target on uncertainties caused by data drift, model discrepancy and environment perturbations in the control process, and finally to realize an optimal and robust dispatch solution. The proposed framework won the championship in CityLearn Challenge 2022, which provided an influential opportunity to investigate the potential of AI application in the energy domain. In addition, comprehensive experiments are conducted to interpret its effectiveness in the real-life scenario of smart building energy management.