Early diagnosis of Alzheimer's Disease (AD), particularly at the mild cognitive impairment stage, is essential for timely intervention. However, this process faces significant barriers, including reliance on subjective assessments and the high cost of advanced imaging techniques. While deep learning offers automated solutions to improve diagnostic accuracy, its widespread adoption remains constrained due to high energy requirements and computational demands, particularly in resource-limited settings. Spiking neural networks (SNNs) provide a promising alternative, as their brain-inspired design is well-suited to model the sparse and event-driven patterns characteristic of neural degeneration in AD. These networks offer the potential for developing interpretable, energy-efficient diagnostic tools. Despite their advantages, existing SNNs often suffer from limited expressiveness and challenges in stable training, which reduce their effectiveness in handling complex medical tasks. To address these shortcomings, we introduce FasterSNN, a hybrid neural architecture that combines biologically inspired Leaky Integrate-and-Fire (LIF) neurons with region-adaptive convolution and multi-scale spiking attention mechanisms. This approach facilitates efficient, sparse processing of 3D MRI data while maintaining high diagnostic accuracy. Experimental results on benchmark datasets reveal that FasterSNN delivers competitive performance with significantly enhanced efficiency and training stability, highlighting its potential for practical application in AD screening. Our source code is available at https://github.com/wuchangw/FasterSNN.