We report on deep Chandra X-ray Telescope imaging observations of 4C 63.20, one of the few known radio galaxies at z>3.5. The X-ray counterpart is resolved into a core plus two off-nuclear sources that (combined) account for close to 30% of the total X-ray flux. Their morphology and orientation are consistent with a diffuse, lobe-like nature, albeit compact hotspots cannot be ruled out. The broadband spectral energy distribution of 4C 63.20 can be reproduced with a jet model where the majority of the radio flux can be ascribed to synchrotron emission from the hotspots, whereas the (non-nuclear) X-ray emission is produced via Inverse Compton (IC) off of Cosmic Microwave Background (CMB) photons within the extended lobes. This scenario is broadly consistent with the expectation from highly magnetized lobes in a hotter CMB, and supports the view that IC/CMB may quench less extreme radio lobes at high redshifts.