Optical coherence tomography (OCT) is a noninvasive technology that enables real-time imaging of tissue microanatomies. The axial resolution of OCT is intrinsically constrained by the spectral bandwidth of the employed light source while maintaining a fixed center wavelength for a specific application. Physically extending this bandwidth faces strong limitations and requires a substantial cost. We present a novel computational approach, called as O-PRESS, for boosting the axial resolution of OCT with Prior Guidance, a Recurrent mechanism, and Equivariant Self-Supervision. Diverging from conventional superresolution methods that rely on physical models or data-driven techniques, our method seamlessly integrates OCT modeling and deep learning, enabling us to achieve real-time axial-resolution enhancement exclusively from measurements without a need for paired images. Our approach solves two primary tasks of resolution enhancement and noise reduction with one treatment. Both tasks are executed in a self-supervised manner, with equivariance imaging and free space priors guiding their respective processes. Experimental evaluations, encompassing both quantitative metrics and visual assessments, consistently verify the efficacy and superiority of our approach, which exhibits performance on par with fully supervised methods. Importantly, the robustness of our model is affirmed, showcasing its dual capability to enhance axial resolution while concurrently improving the signal-to-noise ratio.