We introduce an active-sampling-based framework for automatic prompt optimization, designed to enhance the performance of Large Language Model (LLM)-as-a-judge systems, which use LLMs to evaluate the quality of text or other outputs, in label-scarce settings. Unlike existing approaches that rely on extensive annotations, our method starts with no labeled data and iteratively selects and labels a small, diverse, and informative subset of samples to guide prompt refinement. At each iteration, our method evaluates the current prompt based on selected data and automatically updates the prompt, enabling efficient prompt optimization with minimal supervision. Moreover, we formulate sample selection as a convex optimization problem that balances uncertainty and diversity, maximizing the utility of limited labeling budgets. We validate our framework across four popular LLMs and three real-world datasets, including one from a deployed industry product. Results show that our optimized prompts consistently outperform baselines, achieving significant gains in evaluation quality and robustness while substantially reducing labeling costs.