Enabling robots to learn novel tasks in a data-efficient manner is a long-standing challenge. Common strategies involve carefully leveraging prior experiences, especially transition data collected on related tasks. Although much progress has been made for general pick-and-place manipulation, far fewer studies have investigated contact-rich assembly tasks, where precise control is essential. We introduce SRSA (Skill Retrieval and Skill Adaptation), a novel framework designed to address this problem by utilizing a pre-existing skill library containing policies for diverse assembly tasks. The challenge lies in identifying which skill from the library is most relevant for fine-tuning on a new task. Our key hypothesis is that skills showing higher zero-shot success rates on a new task are better suited for rapid and effective fine-tuning on that task. To this end, we propose to predict the transfer success for all skills in the skill library on a novel task, and then use this prediction to guide the skill retrieval process. We establish a framework that jointly captures features of object geometry, physical dynamics, and expert actions to represent the tasks, allowing us to efficiently learn the transfer success predictor. Extensive experiments demonstrate that SRSA significantly outperforms the leading baseline. When retrieving and fine-tuning skills on unseen tasks, SRSA achieves a 19% relative improvement in success rate, exhibits 2.6x lower standard deviation across random seeds, and requires 2.4x fewer transition samples to reach a satisfactory success rate, compared to the baseline. Furthermore, policies trained with SRSA in simulation achieve a 90% mean success rate when deployed in the real world. Please visit our project webpage https://srsa2024.github.io/.