While Graph Neural Networks (GNNs) have demonstrated their efficacy in dealing with non-Euclidean structural data, they are difficult to be deployed in real applications due to the scalability constraint imposed by the multi-hop data dependency. Existing methods attempt to address this scalability issue by training student multi-layer perceptrons (MLPs) exclusively on node content features using labels derived from the teacher GNNs. However, the trained MLPs are neither effective nor robust. In this paper, we ascribe the lack of effectiveness and robustness to three significant challenges: 1) the misalignment between content feature and label spaces, 2) the strict hard matching to teacher's output, and 3) the sensitivity to node feature noises. To address the challenges, we propose NOSMOG, a novel method to learn NOise-robust Structure-aware MLPs On Graphs, with remarkable effectiveness, robustness, and efficiency. Specifically, we first address the misalignment by complementing node content with position features to capture the graph structural information. We then design an innovative representational similarity distillation strategy to inject soft node similarities into MLPs. Finally, we introduce adversarial feature augmentation to ensure stable learning against feature noises. Extensive experiments and theoretical analyses demonstrate the superiority of NOSMOG by comparing it to GNNs and the state-of-the-art method in both transductive and inductive settings across seven datasets. Codes are available at https://github.com/meettyj/NOSMOG.