In advanced technology nodes, aging effects like negative and positive bias temperature instability (NBTI and PBTI) become increasingly significant, making timing closure and optimization more challenging. Unfortunately, conventional critical path selection tools used in reliability-aware design flow cannot accurately identify critical paths under different aging conditions. To address this issue, we propose an aging-aware critical path selection flow comprising two parts: critical cell detection and path criticality computation. We employ graph-attention networks (GATs) to predict the critical cells in the aged circuits, and a path criticality computation algorithm that takes into account circuit-level and transistor-level parameters to generate path criticality rank lists. Our experimental results demonstrate that our GAT model outperforms classical machine learning models in detecting critical cells. Additionally, compared with the commercial tool, our aging-aware flow achieves an average accuracy of 99.52%, 98.69%, and 97.20% for top-10%, top-5%, and top-1% path sets respectively, in five industrial designs subjected to different aging conditions and workloads.