Movable antennas (MA) have gained significant attention in recent years to overcome the limitations of extremely large antenna arrays in terms of cost and power consumption. In this paper, we investigate the use of MA arrays at the base station (BS) for angle-of-departure (AoD) estimation under uncertainty in the user equipment (UE) location. Specifically, we (i) derive the theoretical performance limits through the Cramér-Rao bound (CRB) and (ii) optimize the antenna positions to ensure robust performance within the UE's uncertainty region. Numerical results show that dynamically optimizing antenna placement by explicitly considering the uncertainty region yields superior performance compared to fixed arrays, demonstrating the ability of MA systems to adapt and outperform conventional arrays.