Graph Neural Networks (GNNs) have garnered considerable interest due to their exceptional performance in a wide range of graph machine learning tasks. Nevertheless, the majority of GNN-based approaches have been examined using well-annotated benchmark datasets, leading to suboptimal performance in real-world graph learning scenarios. To bridge this gap, the present paper investigates the problem of graph transfer learning in the presence of label noise, which transfers knowledge from a noisy source graph to an unlabeled target graph. We introduce a novel technique termed Balance Alignment and Information-aware Examination (ALEX) to address this challenge. ALEX first employs singular value decomposition to generate different views with crucial structural semantics, which help provide robust node representations using graph contrastive learning. To mitigate both label shift and domain shift, we estimate a prior distribution to build subgraphs with balanced label distributions. Building on this foundation, an adversarial domain discriminator is incorporated for the implicit domain alignment of complex multi-modal distributions. Furthermore, we project node representations into a different space, optimizing the mutual information between the projected features and labels. Subsequently, the inconsistency of similarity structures is evaluated to identify noisy samples with potential overfitting. Comprehensive experiments on various benchmark datasets substantiate the outstanding superiority of the proposed ALEX in different settings.