The security-constrained alternating current optimal power flow problem (SC-ACOPF) involves determining the minimum-cost operating conditions for power systems while ensuring system security. This problem is challenging due to its formulation as a large-scale, non-linear, and non-convex model, presenting difficulties for traditional optimization approaches. In this study, we introduce an approximate-and-optimize method to tackle the SC-ACOPF. We decompose the SC-ACOPF model into a two-stage problem formulation, where the first stage addresses the base case, and the second stage focuses on post-contingency decisions. To handle the complexity, we employ a neural network to approximate the objective cost associated with all contingencies. Subsequently, we optimize the first-stage problem after incorporating the neural network surrogate function. Computational experiments conducted on a 500-bus system showcase the effectiveness of our proposed framework. The results demonstrate its superior performance compared to baseline algorithms, achieving a remarkable \(60\%\) reduction in computational time and a \(50\%\) decrease in objective costs.