This paper investigates the efficacy of Convolutional Neural Network (CNN) based methods to navigate autonomously around asteroids. The main contribution of this work is the successful development of a first-of-a-kind pose estimation pipeline, consisting of a CNN-based feature detector and a Perspective-n-Points (PnP) solver to allow accurate, safe, and autonomous distance estimation with respect to a target asteroid. A top-down CNN-based feature detector is developed, consisting of an object and keypoint detection network in sequence, which detects n pre-defined keypoints, designated on the target's 3D model, within the 2D image. The simulated target asteroid is Bennu, a subkilometer asteroid with a spinning top-shape and pronounced equatorial bulge. The networks have been trained and evaluated on synthetic datasets created in this work, consisting of 32,352 images with a variety of poses from 4.5 to 9 km from the asteroid, for different illumination conditions, asteroid orientations, and image corruptions that emulate real sensor artifacts. The pipeline could achieve a mean and median line-of-sight distance estimate of around 42 m and 30 m, respectively, at a confidence level of 90% for the large relative range, while satisfying the accuracy requirement of a maximum of 10% knowledge error for 99.979% of the cases.