The proliferation of dynamic graph-structured data necessitates Continual Graph Learning (CGL) to enable models to learn incrementally and retain past knowledge. Class-incremental learning (class-IL) in CGL is particularly challenging due to catastrophic forgetting. Existing CGL strategies, including recent prompt-based learning methods, often grapple with significant memory overhead and suboptimal alignment between pre-training objectives and downstream continual tasks. We propose TriP, a triple-prompt framework aligning pre-training and class-incremental objectives in continual graph learning. As a lightweight parameter isolation-based method, TriP utilizes feature-level and class-level prompts to precisely capture task-specific knowledge. It introduces a unified prompt template that aligns downstream classification with self-supervised pre-training objectives, which can bridge the semantic gap and maximize the generalization capabilities of pre-trained graph models. Extensive experiments on four public benchmark datasets demonstrate that our method significantly outperforms state-of-the-art approaches, illustrating its effectiveness in mitigating catastrophic forgetting and improving the efficiency of continual learning on graphs.