Monocular depth estimation is challenging due to its inherent ambiguity and ill-posed nature, yet it is quite important to many applications. While recent works achieve limited accuracy by designing increasingly complicated networks to extract features with limited spatial geometric cues from a single RGB image, we intend to introduce spatial cues by training a teacher network that leverages left-right image pairs as inputs and transferring the learned 3D geometry-aware knowledge to the monocular student network. Specifically, we present a novel knowledge distillation framework, named ADU-Depth, with the goal of leveraging the well-trained teacher network to guide the learning of the student network, thus boosting the precise depth estimation with the help of extra spatial scene information. To enable domain adaptation and ensure effective and smooth knowledge transfer from teacher to student, we apply both attention-adapted feature distillation and focal-depth-adapted response distillation in the training stage. In addition, we explicitly model the uncertainty of depth estimation to guide distillation in both feature space and result space to better produce 3D-aware knowledge from monocular observations and thus enhance the learning for hard-to-predict image regions. Our extensive experiments on the real depth estimation datasets KITTI and DrivingStereo demonstrate the effectiveness of the proposed method, which ranked 1st on the challenging KITTI online benchmark.