Rerankers are critical in Retrieval-Augmented Generation (RAG) for filtering evidence that enhances the accurate generation of LLMs. With the extension to open-domain scenarios, rerankers are inevitably deployed on mixed-style corpora, whereas most existing rerankers are mainly trained on well-edited texts. A rarely explored issue lies in enabling rerankers to maximally capture the effective knowledge for downstream LLMs without being misled by stylistic features. To address this issue, we propose SARK (Style-Adaptive Reranker with Knowledge Prioritization), a style-augmented multi-task framework that prioritizes effective knowledge over stylistic perturbations. SARK performs multi-granular knowledge mining by using an LLM to derive passage-level supervision on whether a passage helps or harms answer correctness, and list-level relative ranking preferences over candidate passages. It then jointly optimizes the reranker model with passage-level classification and list-level ranking objectives via style-augmented multi-task learning, encouraging the model to focus on the information needed for answering under mixed-style scenarios. Extensive experiments demonstrate that SARK improves generation performance across multiple LLMs under mixed-style conditions.