Detecting Alzheimer's Disease (AD) from narrative transcripts remains a challenging task for large language models (LLMs), particularly under out-of-distribution (OOD) and data-scarce conditions. While in-context learning (ICL) provides a parameter-efficient alternative to fine-tuning, existing ICL approaches often suffer from task recognition failure, suboptimal demonstration selection, and misalignment between label words and task objectives, issues that are amplified in clinical domains like AD detection. We propose Explicit Knowledge In-Context Learners (EK-ICL), a novel framework that integrates structured explicit knowledge to enhance reasoning stability and task alignment in ICL. EK-ICL incorporates three knowledge components: confidence scores derived from small language models (SLMs) to ground predictions in task-relevant patterns, parsing feature scores to capture structural differences and improve demo selection, and label word replacement to resolve semantic misalignment with LLM priors. In addition, EK-ICL employs a parsing-based retrieval strategy and ensemble prediction to mitigate the effects of semantic homogeneity in AD transcripts. Extensive experiments across three AD datasets demonstrate that EK-ICL significantly outperforms state-of-the-art fine-tuning and ICL baselines. Further analysis reveals that ICL performance in AD detection is highly sensitive to the alignment of label semantics and task-specific context, underscoring the importance of explicit knowledge in clinical reasoning under low-resource conditions.