Dynamic graphs are pervasive in the real world, modeling dynamic relations between objects across various fields. For dynamic graph modeling, dynamic graph neural networks (DGNNs) have emerged as a mainstream technique, which are generally pre-trained on the link prediction task, leaving a significant gap from the objectives of downstream tasks such as node classification. To bridge the gap, prompt-based learning has gained traction on graphs. However, existing efforts focus on static graphs, neglecting the evolution of dynamic graphs. In this paper, we propose DyGPrompt, a novel pre-training and prompting framework for dynamic graph modeling. First, we design dual prompts to address the gap in both task objectives and dynamic variations across pre-training and downstream tasks. Second, we recognize that node and time features mutually characterize each other, and propose dual condition-nets to model the evolving node-time patterns in downstream tasks. Finally, we thoroughly evaluate and analyze DyGPrompt through extensive experiments on three public datasets.