The airborne hyperspectral remote sensing systems (AHRSSs) acquire images with high spectral resolution, high spatial resolution, and high temporal dimension. While the AHRSS captures more detailed information from the terrain objects, the computational complexity of data processing is greatly increased. As an important application technology in the hyperspectral domain, anomaly detection (AD) processing must be real-time and high-precision in many cases, such as post-disaster rescue, military battlefield search, and natural disaster detection. In this paper, the real-time AD technology for the push-broom AHRSS is studied, the mathematical model is established, and a novel implementation framework is proposed. Firstly, the optimized kernel minimum noise fraction (OP-KMNF) transformation is employed to extract informative and discriminative features between the background and anomalies. Secondly, the Nyström method is introduced to reduce the computational complexity of OP-KMNF transformation by decomposing and extrapolating the sub-kernel matrix to estimate the eigenvector of the entire kernel matrix. Thirdly, the extracted features are transferred to hard disks for data storage. Then, taking the extracted features as input data, the background separation model-based CEM anomaly detector (BSM-CEMAD) is imported to detect anomalies. Finally, graphics processing unit (GPU) parallel computing is utilized in the Nyström-based OP-KMNF (NOP-KMNF) transformation and the BSM-CEMAD to improve the execution efficiency, and the real-time AD for the push-broom AHRSS could be realized. To test the feasibility of the implementation framework proposed in this paper, the experiment is carried out with the Airborne Multi-Modular Imaging Spectrometer (AMMIS) developed by the Shanghai Institute of Technical Physics as the data acquisition platform. The experimental results show that the proposed method outperforms many other state-of-the-art AD methods in anomalies detection and background suppression. Moreover, under the condition that the downlink data could retain most of the hyperspectral data information, the proposed method achieves real-time detection of pixel-level anomalies, with the initial delay not exceeding 1 s, the false alarm rate (FAR) less than 5%, and the true positive rate (TPR) close to 98%.