Recently, learning-based robotic navigation systems have gained extensive research attention and made significant progress. However, the diversity of open-world scenarios poses a major challenge for the generalization of such systems to practical scenarios. Specifically, learned systems for scene measurement and state estimation tend to degrade when the application scenarios deviate from the training data, resulting to unreliable depth and pose estimation. Toward addressing this problem, this work aims to develop a visual odometry system that can fast adapt to diverse novel environments in an online manner. To this end, we construct a self-supervised online adaptation framework for monocular visual odometry aided by an online-updated depth estimation module. Firstly, we design a monocular depth estimation network with lightweight refiner modules, which enables efficient online adaptation. Then, we construct an objective for self-supervised learning of the depth estimation module based on the output of the visual odometry system and the contextual semantic information of the scene. Specifically, a sparse depth densification module and a dynamic consistency enhancement module are proposed to leverage camera poses and contextual semantics to generate pseudo-depths and valid masks for the online adaptation. Finally, we demonstrate the robustness and generalization capability of the proposed method in comparison with state-of-the-art learning-based approaches on urban, in-house datasets and a robot platform. Code is publicly available at: https://github.com/jixingwu/SOL-SLAM.