High-fidelity online 3D scene reconstruction from monocular videos continues to be challenging, especially for coherent and fine-grained geometry reconstruction. The previous learning-based online 3D reconstruction approaches with neural implicit representations have shown a promising ability for coherent scene reconstruction, but often fail to consistently reconstruct fine-grained geometric details during online reconstruction. This paper presents a new on-the-fly monocular 3D reconstruction approach, named GP-Recon, to perform high-fidelity online neural 3D reconstruction with fine-grained geometric details. We incorporate geometric prior (GP) into a scene's neural geometry learning to better capture its geometric details and, more importantly, propose an online volume rendering optimization to reconstruct and maintain geometric details during the online reconstruction task. The extensive comparisons with state-of-the-art approaches show that our GP-Recon consistently generates more accurate and complete reconstruction results with much better fine-grained details, both quantitatively and qualitatively.