In the 5G era, the communication networks tend to be ultra-densified, which will improve the accuracy of indoor positioning and further improve the quality of positioning service. In this study, we propose an indoor three-dimensional (3D) dynamic reconstruction fingerprint matching algorithm (DSR-FP) in a 5G ultra-dense network. The first step of the algorithm is to construct a local fingerprint matrix having low-rank characteristics using partial fingerprint data, and then reconstruct the local matrix as a complete fingerprint library using the FPCA reconstruction algorithm. In the second step of the algorithm, a dynamic base station matching strategy is used to screen out the best quality service base stations and multiple sub-optimal service base stations. Then, the fingerprints of the other base station numbers are eliminated from the fingerprint database to simplify the fingerprint database. Finally, the 3D estimated coordinates of the point to be located are obtained through the K-nearest neighbor matching algorithm. The analysis of the simulation results demonstrates that the average relative error between the reconstructed fingerprint database by the DSR-FP algorithm and the original fingerprint database is 1.21%, indicating that the accuracy of the reconstruction fingerprint database is high, and the influence of the location error can be ignored. The positioning error of the DSR-FP algorithm is less than 0.31 m. Furthermore, at the same signal-to-noise ratio, the positioning error of the DSR-FP algorithm is lesser than that of the traditional fingerprint matching algorithm, while its positioning accuracy is higher.