This letter investigates the challenge of channel estimation in a multiuser millimeter-wave (mmWave) time-division duplexing (TDD) system. In this system, the base station (BS) employs a multi-antenna uniform linear array (ULA), while each mobile user is equipped with a fluid antenna system (FAS). Accurate channel state information (CSI) plays a crucial role in the precise placement of antennas in FAS. Traditional channel estimation methods designed for fixed-antenna systems are inadequate due to the high dimensionality of FAS. To address this issue, we propose a low-sample-size sparse channel reconstruction (L3SCR) method, capitalizing on the sparse propagation paths characteristic of mmWave channels. In this approach, each fluid antenna only needs to switch and measure the channel at a few specific locations. By observing this reduced-dimensional data, we can effectively extract angular and gain information related to the sparse channel, enabling us to reconstruct the full CSI. Simulation results demonstrate that our proposed method allows us to obtain precise CSI with minimal hardware switching and pilot overhead. As a result, the system sum-rate approaches the upper bound achievable with perfect CSI.