To meet the greenhouse gas reduction targets and address the uncertainty introduced by the surging penetration of stochastic renewable energy sources, energy storage systems are being deployed in microgrids. Relying solely on short-term uncertainty forecasts can result in substantial costs when making dispatch decisions for a storage system over an entire day. To mitigate this challenge, an adaptive robust optimization approach tailored for a hybrid hydrogen battery energy storage system (HBESS) operating within a microgrid is proposed, with a focus on efficient state-of-charge (SoC) planning to minimize microgrid expenses. The SoC ranges of the battery energy storage (BES) are determined in the day- ahead stage. Concurrently, the power generated by fuel cells and consumed by electrolysis device are optimized. This is followed by the intraday stage, where BES dispatch decisions are made within a predetermined SoC range to accommodate the uncertainties realized. To address this uncertainty and solve the adaptive optimization problem with integer recourse variables in the intraday stage, we proposed an outer-inner column-and-constraint generation algorithm (outer-inner-CCG). Numerical analyses underscored the high effectiveness and efficiency of the proposed adaptive robust operation model in making decisions for HBESS dispatch.