We present molecular dynamics simulation results for solvation dynamics of a simple diatomic solute in model reverse micelles of varying size. These results are compared to solvation dynamics of the probe in spherical cavities of the same size containing only water. Our simulations focus on the short-time dynamics of solvation, from 0 to 2 ps, a significant portion of which has not yet been accessed experimentally. On this time scale, the solvation response in reverse micelles becomes faster as the micelle size parameter, w(0), increases, in agreement with experiment, but most of the effect occurs in the slower, diffusive portion of the response. The short-time inertial dynamics, which account for over 70% of the response in all of the systems studied, appear to be quite robust even when the mobility of individual water molecules is greatly reduced. Decomposition of the nonequilibrium response functions demonstrates that the short time relaxation is dominated by water and occurs at the solute site where hydrogen bonds are broken. Analysis of the equilibrium solvation time correlation functions demonstrates that the linear response approximation is accurate for reverse micelles, but less so for the smooth cavities. Decomposing the equilibrium response into pair and single-molecule contributions, we find that the pair contributions are larger in the reverse micelles and increase as w(0) decreases. This collective response appears to be much faster than the single molecule response and largely offsets the sharp reduction in single molecule mobilities. Another reason for the robustness of the inertial response may be the preferential location of our model probe outside the water layers closest to the interface. The relative magnitudes of fast and slow contributions to the solvent response for a particular chromophore may thus be sensitive to its location relative to the interface.