Graphs have become an important modeling tool for web applications, and Graph Neural Networks (GNNs) have achieved great success in graph representation learning. However, the performance of traditional GNNs heavily relies on a large amount of supervision. Recently, ``pre-train, fine-tune'' has become the paradigm to address the issues of label dependency and poor generalization. However, the pre-training strategies vary for graphs with homophily and heterophily, and the objectives for various downstream tasks also differ. This leads to a gap between pretexts and downstream tasks, resulting in ``negative transfer'' and poor performance. Inspired by prompt learning in Natural Language Processing (NLP), many studies turn to bridge the gap and fully leverage the pre-trained model. However, existing methods for graph prompting are tailored to homophily, neglecting inherent heterophily on graphs. Meanwhile, most of them rely on the randomly initialized prompts, which negatively impact on the stability. Therefore, we propose Self-Prompt, a prompting framework for graphs based on the model and data itself. We first introduce asymmetric graph contrastive learning for pretext to address heterophily and align the objectives of pretext and downstream tasks. Then we reuse the component from pre-training phase as the self adapter and introduce self-prompts based on graph itself for task adaptation. Finally, we conduct extensive experiments on 11 benchmark datasets to demonstrate its superiority. We provide our codes at https://github.com/gongchenghua/Self-Pro.