Employing pre-trained generation models for cross-domain aspect-based sentiment classification has recently led to large improvements. However, they ignore the importance of syntactic structures, which have shown appealing effectiveness in classification based models. Different from previous studies, efficiently encoding the syntactic structure in generation model is challenging because such models are pretrained on natural language, and modeling structured data may lead to catastrophic forgetting of distributional knowledge. In this study, we propose a novel structure-aware generation model to tackle this challenge. In particular, a prompt-driven strategy is designed to bridge the gap between different domains, by capturing implicit syntactic information from the input and output sides. Furthermore, the syntactic structure is explicitly encoded into the structure-aware generation model, which can effectively learn domain-irrelevant features based on syntactic pivot features. Empirical results demonstrate the effectiveness of the proposed structure-aware generation model over several strong baselines. The results also indicate the proposed model is capable of leveraging the input syntactic structure into the generation model.