The frequencies of transitions between hyperfine levels of ground-state atoms can be measured with exquisite precision using magnetic-resonance techniques. This makes hyperfine transitions ideal probes of QED effects originating from the interaction of atoms with the quantized electromagnetic field. One of the most remarkable effects predicted by QED is the Casimir-Polder shift experienced by the energy levels of atoms placed near one or more dielectric objects. Here we compute the Casimir-Polder shift and the width of hyperfine transitions between ground-state Zeeman sublevels of a hydrogen atom placed in a micron-sized metallic cavity, over a range of temperatures extending from cryogenic temperatures to room temperature. Results are presented also for deuterium and tritium. We predict shifts of the hyperfine transitions frequencies of a few tens of Hz that might be measurable with present-day magnetic resonance apparatus.