Massive multiple-input multiple-output (MIMO) radar, enabled by millimeter-wave virtual MIMO techniques, provides great promises to the high-resolution automotive sensing and target detection in unmanned ground/aerial vehicles (UGA/UAV). As a long-established problem, however, existing subspace methods suffer from either high complexity or low accuracy. In this work, we propose two efficient methods, to accomplish fast subspace computation and accurate angle of arrival (AoA) acquisition. By leveraging randomized low-rank approximation, our fast multiple signal classification (MUSIC) methods, relying on random sampling and projection techniques, substantially accelerate the subspace estimation by orders of magnitude. Moreover, we establish the theoretical bounds of our proposed methods, which ensure the accuracy of the approximated pseudo-spectrum. As demonstrated, the pseudo-spectrum acquired by our fast-MUSIC would be highly precise; and the estimated AoA is almost as accurate as standard MUSIC. In contrast, our new methods are tremendously faster than standard MUSIC. Thus, our fast-MUSIC enables the high-resolution real-time environmental sensing with massive MIMO radars, which has great potential in the emerging unmanned systems.