This paper explores how a flying drone can autonomously navigate while constructing a narrowband radio map for signal localization. As flying drones become more ubiquitous, their wireless signals will necessitate new wireless technologies and algorithms to provide robust radio infrastructure while preserving radio spectrum usage. A potential solution for this spectrum-sharing localization challenge is to limit the bandwidth of any transmitter beacon. However, location signaling with a narrow bandwidth necessitates improving a wireless aerial system's ability to filter a noisy signal, estimate the transmitter's location, and self-pilot to improve the location estimate. By showing results through simulation, emulation, and a final drone flight experiment, this work provides an algorithm using a Gaussian process for radio signal estimation and Bayesian optimization for drone automatic guidance. This research supports advanced radio and aerial robotics applications in critical areas such as search-and-rescue, last-mile delivery, and large-scale platform digital twin development.