Unmanned aerial vehicle (UAV) is considered as an important component for future communication networks. In this paper, an air-to-ground (A2G) channel sounder is designed and implemented for UAV communication channel measurement and characterization. The channel impulse response (CIR) extraction is implemented on a field programmable gate array (FPGA) to improve extraction efficiency. Based on the channel characteristics under measured (or baseline) scenarios, a transfer learning neural network (TNN) framework is also proposed to predict the channel characteristics of other unmeasured (or transferred) scenarios. In the proposed framework, the baseline matrices of neural network parameters are obtained from the measurement data of baseline scenarios. The ray tracing (RT) simulation data is only used to obtain the extrapolation matrices where we utilize imperfect digital map and do not require a highly accurate RT simulation. Then the neural network driven by the baseline and extrapolation matrices is used to predict the channel characteristics of transferred scenarios. To verify the proposed prediction method, the channel characteristics including path loss, K-factor, and root mean square delay spread of a near-urban scenario are firstly measured. Then, the corresponding channel characteristics of a transferred dense-urban scenario are predicted by the proposed TNN method and validated by the measurement data. It is shown that the predicted channel characteristics are well consistent with the measured ones.