Aspect Category Sentiment Analysis (ACSA), which aims to identify fine-grained sentiment polarities of the aspect categories discussed in user reviews. ACSA is challenging and costly when conducting it into real-world applications, that mainly due to the following reasons: 1.) Labeling the fine-grained ACSA data is often labor-intensive. 2.) The aspect categories will be dynamically updated and adjusted with the development of application scenarios, which means that the data must be relabeled frequently. 3.) Due to the increase of aspect categories, the model must be retrained frequently to fast adapt to the newly added aspect category data. To overcome the above-mentioned problems, we introduce a novel Meta Multi-Task Learning (MMTL) approach, that frame ACSA tasks as a meta-learning problem (i.e., regarding aspect-category sentiment polarity classification problems as the different training tasks for meta-learning) to learn an ideal and shareable initialization for the multi-task learning model that can be adapted to new ACSA tasks efficiently and effectively. Experiment results show that the proposed approach significantly outperforms the strong pre-trained transformer-based baseline model, especially, in the case of less labeled fine-grained training data.