Contextual decision-making problems have witnessed extensive applications in various fields such as online content recommendation, personalized healthcare, and autonomous vehicles, where a core practical challenge is to select a suitable surrogate model for capturing unknown complicated reward functions. It is often the case that both high approximation accuracy and explicit uncertainty quantification are desired. In this work, we propose a neural network-accompanied Gaussian process (NN-AGP) model, which leverages neural networks to approximate the unknown and potentially complicated reward function regarding the contextual variable, and maintains a Gaussian process surrogate model with respect to the decision variable. Our model is shown to outperform existing approaches by offering better approximation accuracy thanks to the use of neural networks and possessing explicit uncertainty quantification from the Gaussian process. We also analyze the maximum information gain of the NN-AGP model and prove regret bounds for the corresponding algorithms. Moreover, we conduct experiments on both synthetic and practical problems, illustrating the effectiveness of our approach.