Temporal Graph Neural Networks (TGNNs) lay emphasis on capturing node interactions over time but often overlook evolution in node classes and dynamic data distributions triggered by the continuous emergence of new class labels, known as the open-set problem. This problem poses challenges for existing TGNNs in preserving learned classes while rapidly adapting to new, unseen classes. To address this, this paper identifies two primary factors affecting model performance on the open temporal graph, backed by a theoretical guarantee: (1) the forgetting of prior knowledge and (2) distribution discrepancies between successive tasks. Building on these insights, we propose N-ForGOT, which incorporates two plug-in modules into TGNNs to preserve prior knowledge and enhance model generalizability for new classes simultaneously. The first module preserves previously established inter-class connectivity and decision boundaries during the training of new classes to mitigate the forgetting caused by temporal evolutions of class characteristics. The second module introduces an efficient method for measuring distribution discrepancies with designed temporal Weisfeiler-Lehman subtree patterns, effectively addressing both structural and temporal shifts while reducing time complexity. Experimental results on four public datasets demonstrate that our method significantly outperforms state-of-the-art approaches in prediction accuracy, prevention of forgetting, and generalizability.