As VR devices become increasingly prevalent, live 360-degree video has surged in popularity. However, current live 360-degree video systems heavily rely on uplink bandwidth to deliver high-quality live videos. Recent advancements in neural-enhanced streaming offer a promising solution to this limitation by leveraging server-side computation to conserve bandwidth. Nevertheless, these methods have primarily concentrated on neural enhancement within a single domain (either spatial or temporal), which may not adeptly adapt to diverse video scenarios and fluctuating bandwidth conditions. In this paper, we propose Lumos, a novel spatial-temporal integrated neural-enhanced live 360-degree video streaming system. To accommodate varied video scenarios, we devise a real-time Neural-enhanced Quality Prediction (NQP) model to predict the neural-enhanced quality for different video contents. To cope with varying bandwidth conditions, we design a Content-aware Bitrate Allocator, which dynamically allocates bitrates and selects an appropriate neural enhancement configuration based on the current bandwidth. Moreover, Lumos employs online learning to improve prediction performance and adjust resource utilization to optimize user quality of experience (QoE). Experimental results demonstrate that Lumos surpasses state-of-the-art neural-enhanced systems with an improvement of up to 0.022 in terms of SSIM, translating to an 8.2%-8.5% enhancement in QoE for live stream viewers.