This paper addresses the important yet underexplored task of multi-class sentiment analysis (MCSA), which remains challenging due to subtle semantic differences between adjacent sentiment categories and the scarcity of high-quality annotated data. To tackle these challenges, RD-MCSA (Rationales and Demonstrations-based Multi-Class Sentiment Analysis) is proposed as an In-Context Learning (ICL) framework designed to improve MCSA performance under limited supervision by integrating classification rationales and adaptively selected demonstrations. First, semantically grounded classification rationales are generated from a representative, class-balanced subset of annotated samples selected using a tailored balanced coreset algorithm. These rationales are then paired with demonstrations selected via a similarity-based mechanism powered by a multi-kernel Gaussian process (MK-GP), enabling large language models (LLMs) to better capture fine-grained sentiment distinctions. Experiments on five benchmark sentiment datasets show that RD-MCSA consistently outperforms both supervised baselines and standard ICL methods across various evaluation metrics.