A neo-neutron star is a hot neutron star that has just become transparent to neutrinos. In a core-collapse supernova or accretion-induced collapse of a white dwarf, the neo-neutron star phase directly follows the proto-neutron star phase, about 30-60 s after the initial collapse. It will also be present in a binary neutron star merger in the case where the "born-again" hot massive compact star does not immediately collapse into a black hole. Eddington or even super-Eddington luminosities are present for some time. A neo-neutron star produced in a core-collapse supernova is not directly observable, but the one produced by a binary merger, likely associated with an off-axis short gamma-ray burst, may be observable for some time as well as when produced in the accretion-induced collapse of a white dwarf. We present a first step in the study of this neo-neutron star phase in a spherically symmetric configuration, thus ignoring fast rotation and also ignoring the effect of strong magnetic fields. We put particular emphasis on determining how long the star can sustain a near-Eddington luminosity and also show the importance of positrons and contraction energy during the neo-neutron star phase. We finally discuss the observational prospects for neutron star mergers triggered by LIGO and for accretion-induced collapse transients.