Analysis and visualization of 3D microscopy images pose challenges due to anisotropic axial resolution, demanding volumetric super-resolution along the axial direction. While training a learning-based 3D super-resolution model seems to be a straightforward solution, it requires ground truth isotropic volumes and suffers from the curse of dimensionality. Therefore, existing methods utilize 2D neural networks to reconstruct each axial slice, eventually piecing together the entire volume. However, reconstructing each slice in the pixel domain fails to give consistent reconstruction in all directions leading to misalignment artifacts. In this work, we present a reconstruction framework based on implicit neural representation (INR), which allows 3D coherency even when optimized by independent axial slices in a batch-wise manner. Our method optimizes a continuous volumetric representation from low-resolution axial slices, using a 2D diffusion prior trained on high-resolution lateral slices without requiring isotropic volumes. Through experiments on real and synthetic anisotropic microscopy images, we demonstrate that our method surpasses other state-of-the-art reconstruction methods. The source code is available on GitHub: https://github.com/hvcl/INR-diffusion.