The hydrodynamics around a Brownian particle has a noticeable impact on its hindered diffusion in arbitrary geometries (such as channels/pores) due to reduced mobility close to walls. These effects are difficult to describe at sub-pore scales, wherein a complete analytical solution of the underlying hydrodynamics is challenging to obtain. Here, we propose a coupled Langevin-multiphase direct numerical simulation (DNS) framework, that fully resolves the hydrodynamics in such systems and consequently provides an on-the-fly capability to probe local instantaneous particle diffusivities. We validate and establish the capabilities of this framework in square micro-channels (under varying degrees of hydrodynamic confinement) and in an arbitrary pore. Our results show that directional variations in mean-squared displacements, velocity auto-correlation functions and diffusivities of the Brownian particle, due to inherent asymmetries in the geometry are adequately captured. Further, a local anisotropy in the hydrodynamic resistances along the co-axial direction of the channel is also noted.