Accurate building segmentation and height estimation from single-view RGB satellite imagery are fundamental for urban analytics, yet remain ill-posed due to structural variability and the high computational cost of global context modeling. While current approaches typically adapt monocular depth architectures, they often suffer from boundary bleeding and systematic underestimation of high-rise structures. To address these limitations, we propose BuildMamba, a unified multi-task framework designed to exploit the linear-time global modeling of visual state-space models. Motivated by the need for stronger structural coupling and computational efficiency, we introduce three modules: a Mamba Attention Module for dynamic spatial recalibration, a Spatial-Aware Mamba-FPN for multi-scale feature aggregation via gated state-space scans, and a Mask-Aware Height Refinement module using semantic priors to suppress height artifacts. Extensive experiments demonstrate that BuildMamba establishes a new performance upper bound across three benchmarks. Specifically, it achieves an IoU of 0.93 and RMSE of 1.77~m on DFC23 benchmark, surpassing state-of-the-art by 0.82~m in height estimation. Simulation results confirm the model's superior robustness and scalability for large-scale 3D urban reconstruction.