Recent advancements in Large Language Models (LLMs) have shown significant potential in enhancing recommender systems. However, addressing the cold-start recommendation problem, where users lack historical data, remains a considerable challenge. In this paper, we introduce KALM4Rec (Keyword-driven Retrieval-Augmented Large Language Models for Cold-start User Recommendations), a novel framework specifically designed to tackle this problem by requiring only a few input keywords from users in a practical scenario of cold-start user restaurant recommendations. KALM4Rec operates in two main stages: candidates retrieval and LLM-based candidates re-ranking. In the first stage, keyword-driven retrieval models are used to identify potential candidates, addressing LLMs' limitations in processing extensive tokens and reducing the risk of generating misleading information. In the second stage, we employ LLMs with various prompting strategies, including zero-shot and few-shot techniques, to re-rank these candidates by integrating multiple examples directly into the LLM prompts. Our evaluation, using a Yelp restaurant dataset with user reviews from three English-speaking cities, shows that our proposed framework significantly improves recommendation quality. Specifically, the integration of in-context instructions with LLMs for re-ranking markedly enhances the performance of the cold-start user recommender system.