Object localization in 3D space is a challenging aspect in monocular 3D object detection. Recent advances in 6DoF pose estimation have shown that predicting dense 2D-3D correspondence maps between image and object 3D model and then estimating object pose via Perspective-n-Point (PnP) algorithm can achieve remarkable localization accuracy. Yet these methods rely on training with ground truth of object geometry, which is difficult to acquire in real outdoor scenes. To address this issue, we propose MonoRUn, a novel detection framework that learns dense correspondences and geometry in a self-supervised manner, with simple 3D bounding box annotations. To regress the pixel-related 3D object coordinates, we employ a regional reconstruction network with uncertainty awareness. For self-supervised training, the predicted 3D coordinates are projected back to the image plane. A Robust KL loss is proposed to minimize the uncertainty-weighted reprojection error. During testing phase, we exploit the network uncertainty by propagating it through all downstream modules. More specifically, the uncertainty-driven PnP algorithm is leveraged to estimate object pose and its covariance. Extensive experiments demonstrate that our proposed approach outperforms current state-of-the-art methods on KITTI benchmark.