Developing a universal graph model capable of generalizing across diverse graph domains has consistently been a key objective in graph learning. Recently, many studies have focused on achieving in-context learning (ICL) on graphs, which can generalize to novel tasks without the need for fine-tuning, similar to large language models (LLMs) such as GPT-3. These researches can be primarily divided into graph-based methods and LLM-based methods. However, the generalization performance of the former is limited by the representation capability of GNNs, while the latter faces the challenge of LLMs understanding graph structures. Therefore, we propose CAGML, a context-aware graph meta-learning model, which learns to generalize to cross-domain and cross-granularity graph tasks using a meta-trained Transformer. Firstly, we formulate graph few-shot learning tasks as a structure-aware sequence modeling problem to unify cross-domain and cross-granularity tasks. Then, a structure-aware Transformer (SAT) is introduced as a graph in-context learner to make predictions with a few labels and the task-specific structural context. Finally, we pre-train SAT in a meta-optimization manner on large-scale citation network and knowledge graph. Experiments on 6 cross-domain graph datasets show that, without fine-tuning, CAGML can achieve state-of-the-art (SOTA) performance in terms of average performance across cross-granularity tasks on adopted datasets.