The 5G cellular network provides a sufficient data rate to support the ever growing number of devices that will be connected to the Internet. To enable Internet of Things (IoT) applications in remote locations, it is very crucial to increase the coverage of the cellular networks. This article proposes a wireless communication system that exploits a full-duplex enabled unmanned aerial vehicle (FD-UAV) integrated into a communication network as a mobile relay to expand network coverage. With the deployment of IoT devices, a large volume of data is introduced into the network, causing a delay in data delivery due to scarcity of resources. Real-time information updates play an important role in modern IoT applications including connected vehicles in intelligent transportation systems, digital healthcare, real-time tracking, and so on. Decision making based on the latest fresh information is critically important for these applications. We quantify the freshness of data in IoT applications using a new performance metric, the age of information (AoI). This work proposes a new mathematical model to estimate the average AoI (AAoI) in the FD-UAV assisted cooperative wireless communication system. Here, sensors located in a transport infrastructure harvest energy from radio frequency (RF) signals transmitted by the FD-UAV via a wireless power transmission RF energy harvesting (EH) technique. Then sensors consume this energy to transmit their sensed real-time observations to the access point with the aid of FD-UAV. A closed-form expression for the AAoI is derived as a function of time allocated for energy harvesting. The optimal time allocation for EH that minimizes the AAoI is identified together with the effect of self-interference experienced by the UAV.