Vision Transformers (ViTs) have achieved remarkable success across various vision tasks, yet their deployment is often hindered by prohibitive computational costs. While structured weight pruning and token compression have emerged as promising solutions, they suffer from prolonged retraining times and global propagation that creates optimization challenges, respectively. We propose ToaSt, a decoupled framework applying specialized strategies to distinct ViT components. We apply coupled head-wise structured pruning to Multi-Head Self-Attention modules, leveraging attention operation characteristics to enhance robustness. For Feed-Forward Networks (over 60\% of FLOPs), we introduce Token Channel Selection (TCS) that enhances compression ratios while avoiding global propagation issues. Our analysis reveals TCS effectively filters redundant noise during selection. Extensive evaluations across nine diverse models, including DeiT, ViT-MAE, and Swin Transformer, demonstrate that ToaSt achieves superior trade-offs between accuracy and efficiency, consistently outperforming existing baselines. On ViT-MAE-Huge, ToaSt achieves 88.52\% accuracy (+1.64 \%) with 39.4\% FLOPs reduction. ToaSt transfers effectively to downstream tasks, cccccachieving 52.2 versus 51.9 mAP on COCO object detection. Code and models will be released upon acceptance.