We investigate the quantum electrodynamic (QED) properties of an atomic electron close to the focus of a spherical mirror. We first show that the spontaneous emission and excited-state level shift of the atom can be fully suppressed with mirror-atom distances of many wavelengths. A three-dimensional theory predicts that the spectral density of vacuum fluctuations can indeed vanish within a volume λ3 around the atom, with the use of a far-distant mirror covering only half of the atomic emission solid angle. The modification of these QED atomic properties is also computed as a function of the mirror size, and large effects are found for only moderate numerical apertures. We also evaluate the long-distance ground-state energy shift (Casimir-Polder shift) and find that it scales as (λ/R)2 at the focus of a hemispherical mirror of radius R, as opposed to the well-known (λ/R)4 scaling law for an atom at a distance R from an infinite plane mirror. Our results are relevant for investigations of QED effects as well as free-space coupling to single atoms using high-numerical-aperture lenses.