In online advertising, advertisers commonly utilize auto-bidding services to bid for impression opportunities. A typical objective of the auto-bidder is to optimize the advertiser's cumulative value of winning impressions within specified budget constraints. However, such a problem is challenging due to the complex bidding environment faced by diverse advertisers. To address this challenge, we introduce ABPlanner, a few-shot adaptable budget planner designed to improve budget-constrained auto-bidding. ABPlanner is based on a hierarchical bidding framework that decomposes the bidding process into shorter, manageable stages. Within this framework, ABPlanner allocates the budget across all stages, allowing a low-level auto-bidder to bids based on the budget allocation plan. The adaptability of ABPlanner is achieved through a sequential decision-making approach, inspired by in-context reinforcement learning. For each advertiser, ABPlanner adjusts the budget allocation plan episode by episode, using data from previous episodes as prompt for current decisions. This enables ABPlanner to quickly adapt to different advertisers with few-shot data, providing a sample-efficient solution. Extensive simulation experiments and real-world A/B testing validate the effectiveness of ABPlanner, demonstrating its capability to enhance the cumulative value achieved by auto-bidders.