With the dynamic air traffic demand and the constrained capacity resources, accurately predicting airport throughput is essential to ensure the efficiency and resilience of air traffic operations. Many research efforts have been made to predict traffic throughputs or flight delays at an airport or over a network. However, it is still a challenging problem due to the complex spatiotemporal dynamics of the highly interacted air transportation systems. To address this challenge, we propose a novel deep learning model, graph attention neural network stacking with a Long short-term memory unit (GAT-LSTM), to predict the short-term airport throughput over a national air traffic network. LSTM layers are included to extract the temporal correlations in the data, while the graph attention mechanism is used to capture the spatial dependencies. For the graph attention mechanism, two graph modeling methods, airport-based graph and OD-pair graph are explored in this study. We tested the proposed model using real-world air traffic data involving 65 major airports in China over 3 months in 2017 and compared its performance with other state-of-the-art models. Results showed that the temporal pattern was the dominate factor, compared to the spatial pattern, in predicting airport throughputs over an air traffic network. Among the prediction models that we compared, both the proposed model and LSTM performed well on prediction accuracy over the entire network. Better performance of the proposed model was observed when focusing on airports with larger throughputs. We also conducted an analysis on model interpretability. We found that spatiotemporal correlations in the data were learned and shown via the model parameters, which helped us to gain insights into the topology and the dynamics of the air traffic network.