Unmanned air vehicles (UAVs) have traditionally been considered as "eyes in the sky", that can move in three dimensions and need to avoid any contact with their environment. On the contrary, contact should not be considered as a problem, but as an opportunity to expand the range of UAVs applications. In this paper, we designed, fabricated, and characterized a whisker sensor unit based on MEMS barometers suitable for tactile localization on UAVs, featuring lightweight, low stiffness, high sensitivity, a broad sensing range, and scalability. Then, for the challenging task of contact point localization, we propose a Recurrent Multi-output Network (RMN) for predicting 3D contact points under continuous contact conditions to address the problems of non-linearity, hysteresis, and non-injective mapping between signals and contact points by considering time series. In addition, we propose an azimuth prediction loss function which reduces the RMSE by 3.24° compared to L₁ loss. Finally, we conduct experiments on a linear stage to validate the 3D contact point localization capability of the proposed whisker system and model. The results show that our localization can achieve excellent performance, with an inference time of 1.4 ms and a mean error of only 9.18 mm in Euclidean distance within 3D space, laying a robust foundation for future implementation of tactile localization on UAVs. The design files, dataset, and source code are available on: https://github.com/BioMorphic-Intelligence-Lab/Whisker-3D-Localization.