Despite remarkable success in diverse web-based applications, Graph Neural Networks(GNNs) inherit and further exacerbate historical discrimination and social stereotypes, which critically hinder their deployments in high-stake domains such as online clinical diagnosis, financial crediting, etc. However, current fairness research that primarily craft on i.i.d data, cannot be trivially replicated to non-i.i.d. graph structures with topological dependence among samples. Existing fair graph learning typically favors pairwise constraints to achieve fairness but fails to cast off dimensional limitations and generalize them into multiple sensitive attributes; besides, most studies focus on in-processing techniques to enforce and calibrate fairness, constructing a model-agnostic debiasing GNN framework at the pre-processing stage to prevent downstream misuses and improve training reliability is still largely under-explored. Furthermore, previous work on GNNs tend to enhance either fairness or privacy individually but few probe into their interplays. In this paper, we propose a novel model-agnostic debiasing framework named MAPPING (\underline{M}asking \underline{A}nd \underline{P}runing and Message-\underline{P}assing train\underline{ING}) for fair node classification, in which we adopt the distance covariance($dCov$)-based fairness constraints to simultaneously reduce feature and topology biases in arbitrary dimensions, and combine them with adversarial debiasing to confine the risks of attribute inference attacks. Experiments on real-world datasets with different GNN variants demonstrate the effectiveness and flexibility of MAPPING. Our results show that MAPPING can achieve better trade-offs between utility and fairness, and mitigate privacy risks of sensitive information leakage.