Subgraph query is a critical task in graph analysis with a wide range of applications across various domains. Most existing methods rely on heuristic vertex matching orderings, which may significantly degrade enumeration performance for certain queries. While learning-based optimizers have recently gained attention in the context of relational databases, they cannot be directly applied to subgraph queries due to the heterogeneous and schema-flexible nature of graph data, as well as the large number of joins involved in subgraph queries. These complexities often leads to inefficient online performance, making such approaches impractical for real-world graph database systems. To address this challenge, we propose NeuSO, a novel learning-based optimizer for subgraph queries that achieves both high accuracy and efficiency. NeuSO features an efficient query graph encoder and an estimator which are trained using a multi-task framework to estimate both subquery cardinality and execution cost. Based on these estimates, NeuSO employs a top-down plan enumerator to generate high-quality execution plans for subgraph queries. Extensive experiments on multiple datasets demonstrate that NeuSO outperforms existing subgraph query ordering approaches in both performance and efficiency.