Precise Event Spotting aims to localize fast-paced actions or events in videos with high temporal precision, a key task for applications in sports analytics, robotics, and autonomous systems. Existing methods typically process all frames uniformly, overlooking the inherent spatio-temporal redundancy in video data. This leads to redundant computation on non-informative regions while limiting overall efficiency. To remain tractable, they often spatially downsample inputs, losing fine-grained details crucial for precise localization. To address these limitations, we propose \textbf{AdaSpot}, a simple yet effective framework that processes low-resolution videos to extract global task-relevant features while adaptively selecting the most informative region-of-interest in each frame for high-resolution processing. The selection is performed via an unsupervised, task-aware strategy that maintains spatio-temporal consistency across frames and avoids the training instability of learnable alternatives. This design preserves essential fine-grained visual cues with a marginal computational overhead compared to low-resolution-only baselines, while remaining far more efficient than uniform high-resolution processing. Experiments on standard PES benchmarks demonstrate that \textbf{AdaSpot} achieves state-of-the-art performance under strict evaluation metrics (\eg, $+3.96$ and $+2.26$ mAP$@0$ frames on Tennis and FineDiving), while also maintaining strong results under looser metrics. Code is available at: \href{https://github.com/arturxe2/AdaSpot}{https://github.com/arturxe2/AdaSpot}.