Understanding and quantifying node importance is a fundamental problem in network science and engineering, underpinning a wide range of applications such as influence maximization, social recommendation, and network dismantling. Prior research often relies on centrality measures or advanced graph embedding techniques using structural information, followed by downstream classification or regression tasks to identify critical nodes. However, these approaches typically decouple node representation learning from the ranking objective and depend heavily on the topological structure of target networks, leading to feature-task inconsistency and poor cross-network generalization. This paper proposes a novel framework that leverages causal representation learning to obtain robust and invariant node embeddings for cross-network ranking tasks. Specifically, we introduce an influence-aware causal node embedding module within an autoencoder architecture to extract node embeddings that are causally related to node importance. Furthermore, we design a unified optimization framework incorporating a causal ranking loss that jointly optimizes reconstruction and ranking objectives, thereby enabling mutual reinforcement between node representation learning and ranking optimization. This design allows the proposed model to be trained on synthetic networks and to generalize effectively across diverse real-world networks. Extensive experiments on multiple benchmark datasets demonstrate that the proposed model consistently outperforms state-of-the-art baselines in terms of both ranking accuracy and cross-network transferability, offering new insights for network analysis and engineering applications-particularly in scenarios where the target network's structure is inaccessible in advance due to privacy or security constraints.