Cellular systems are undergoing a transformation toward the fifth generation (5G). Envisioned applications in 5G include intelligent transport system (ITS), autonomous vehicles, and robots as a part of future roads, factories, and society. These applications rely to a great extent on accurate and timely location information of connected devices. This paper proposes a practical scheme for acquiring precise and timely position information by means of a user-centric ultra-dense network (UDN) architecture based on an edge cloud. The considered solution consists of estimating and tracking the azimuth angle-of-arrival (AoA) of the line-of-sight (LoS)-path between a device and multiple transmission-reception points (TRPs), each having a uniform linear antenna array (ULA). AoA estimates from multiple TRPs are fused into position estimates at the edge cloud to obtain timely position information. The extensive measurements have been carried out using a proof-of-concept software-defined-radio (SDR) testbed in order to experimentally assess the achievable positioning accuracy of the proposed architecture. A realistic UDN deployment scenario has been considered in which TRPs consist of antenna arrays mounted on lamp posts. Our results show that practical UDNs can provide sub-meter positioning accuracy of mobile users by employing ULAs with at least four antennas per TRP and by taking into account the non-idealities of the ULAs' phase and magnitude response.