LLM-as-a-Judge has emerged as a scalable alternative to costly human evaluation. However, LLM judges suffer from position bias in pairwise evaluation: the judge systematically favors the first or second candidate regardless of content, forcing practitioners to perform bidirectional evaluation at twice the inference cost. We propose Probe-and-Steer (PaS), a probe-localized, single-layer steering method for mitigating position bias. PaS localizes the bias-critical layer via a layer-wise logit-difference probe and trains a single residual-stream steering vector at that layer under a multi-objective loss, requiring minimal labeled data while keeping the LLM fully frozen. Through layer-wise probing across multiple model families, we uncover three qualitatively distinct bias-distribution patterns and delineate a clear applicability boundary for residual-stream steering. Experiments on an XSum-based dataset with Llama-3.1-8B-Instruct show that PaS delivers the largest gains among all compared methods, improving consistency by 22.9 pp and accuracy by 16.4 pp while shrinking positional asymmetry by 33× (from 0.471 to 0.014), adding the lowest inference overhead at 4%.