The dynamics of anisotropic nano- to micro scale colloidal particles in confined environments, either near neighboring particles or boundaries, is relevant to a wide range of applications. We utilized Brownian dynamics simulations to predict the translational and rotational fluctuations of a Janus sphere with a cap of nonmatching density near a boundary. The presence of the cap significantly impacted the rotational dynamics of the particle as a consequence of gravitational torque at experimentally relevant conditions. Gravitational torque dominated stochastic torque for a particle >1 μm in diameter and with a 20-nm-thick gold cap. Janus particles at these conditions sampled mostly cap-down or "quenched" orientations. Although the results summarized herein showed that particles of smaller diameter (<1 μm) with a thin gold coating (<5 nm) behave similarly to an isotropic particle, small increases in either particle diameter or coating thickness quenched the polar rotation of the particle. Histogram landscapes of the separation distance from the boundary and orientation observations of particles with larger diameters or thicker gold coatings were mostly populated with quenched configurations. Finally, the histogram landscapes were inverted to obtain the potential energy landscapes, providing a road map for experimental data to be interpreted.