With the advantages of satisfying service requirements and providing high security, standalone private fifth generation (5G) network is perceived as a promising technology for vertical industries. However, to manage the cloud-native network functions (CNFs) in an effective manner, a sophisticated control plane management scheme should be designed in standalone private 5G networks. In this paper, we propose a deep Q-network based CNF placement algorithm (DQN-CNFPA), that jointly minimizes the cost occurred in launching and operating CNFs on edge clouds and the back-haul control traffic overhead. In addition, DQN-CNFPA learns spatiotemporal patterns in service requests and places CNFs in consideration of future cost leveraged by the previous CNF placement strategy. Evaluation results demonstrate that DQN-CNFPA can reduce the cost per hour up to 11.2% compared to the scheme without learning spatiotemporal service request patterns.