Objective. Reliable, continuous neural sensing on wearable edge platforms is fundamental to long-term health monitoring; however, for electroencephalography (EEG)-based sleep monitoring, dense high-frequency processing is often computationally prohibitive under tight energy budgets. Approach. To address this bottleneck, this paper proposes NeuroSleep, an integrated event-driven sensing and inference system for energy-efficient sleep staging. NeuroSleep first converts raw EEG into complementary multi-scale bipolar event streams using Residual Adaptive Multi-Scale Delta Modulation (R-AMSDM), enabling an explicit fidelity-sparsity trade-off at the sensing front end. Furthermore, NeuroSleep adopts a hierarchical inference architecture that comprises an Event-based Adaptive Multi-scale Response (EAMR) module for local feature extraction, a Local Temporal-Attention Module (LTAM) for context aggregation, and an Epoch-Leaky Integrate-and-Fire (ELIF) module to capture long-term state persistence. Main results. Experimental results using subject-independent 5-fold cross-validation on the Sleep-EDF Expanded sleep-cassette (SC) subset with single-channel EEG demonstrate that NeuroSleep achieves a mean accuracy of 74.2% with only 0.932 M parameters while reducing sparsity-adjusted effective operations by approximately 53.6% relative to dense processing. Compared to the representative dense Transformer baseline, NeuroSleep improves accuracy by 7.5% with a 45.8% reduction in computational load. Significance. By coupling neuromorphic event encoding with state-aware context modeling, NeuroSleep offers a deployment-oriented framework for single-channel sleep staging that reduces redundant high-rate processing and improves energy scalability for wearable and edge platforms.