With the rapid development and broad applications of unmanned aerial vehicle (UAV) based wireless stations in the sky, fundamental understanding and characterization of the realistic air-to-ground (A2G) communication link properties are crucial. In this paper, a UAV-aided channel sounder with a real- time processing hardware system is developed for highly dynamic and non-stationary A2G channel measurements. In the hardware system, a global positioning system (GPS) -based triggering signal is designed, the equivalent antenna pattern affected by UAV airframe is considered, and an appropriate sounding signal is selected, in order to improve the accuracy of measured channel impulse response (CIR). Moreover, real-time hardware processing algorithms for raw channel data, i.e., CIR extraction, system response elimination (SRE), power loss recovery (PLR), and adaptive multi-path component (MPC) recognition are developed and implemented on a single field-programmable gate array (FPGA) chip. In this way, the required storage size of channel data and the processing time for one slice of CIR is greatly decreased, which can meet the requirement of non-stationary A2G channel measurement with a high sampling rate and long- time measurement. A commercial channel emulator is used to reproduce controllable channels and verify the performance of developed channel sounder. Finally, the developed channel sounder is applied to carry out A2G measurement campaigns at 3.5 GHz in a campus scenario. The channel characteristics, i.e., path loss (PL), K-factor, and path angle are analyzed. The measured channel characteristics are consistent with existing measurements under a similar scenario. The estimated path angles are also validated by the theoretical results. Thus, the channel sounder can be used to capture the non-stationary A2G channel characteristics for the system design and algorithm optimization of A2G communications.