Single-particle localization of an ultracold atom is studied in one dimension when the atom is confined by an optical lattice and by the incommensurate potential of a high-finesse optical cavity. In the strong-coupling regime the atom is a dynamical refractive medium, the cavity resonance depends on the atomic position within the standing-wave mode, and nonlinearly determines the depth and form of the incommensurate potential. We show that the particular form of the quasirandom cavity potential leads to the appearance of mobility edges, even in presence of nearest-neighbor hopping. We provide a detailed characterization of the system as a function of its parameters and, in particular, of the strength of the atom-cavity coupling, which controls the functional form of the cavity potential. For strong atom-photon coupling the properties of the mobility edges significantly depend on the ratio between the periodicities of the confining optical lattice and of the cavity field.