We study the incidence of nuclear activity in a large sample of massive post-starburst galaxies at z~0.7 selected from the Sloan Digital Sky Survey, and identify active galactic nuclei based on radio continuum and optical emission lines. Over our mass range of 10^10.6-10^11.5 Msun, the incidence of radio activity is weakly dependent on stellar mass and independent of stellar age, while radio luminosity depends strongly on stellar mass. Optical nuclear activity incidence depends most strongly on the Dn4000 line index, a proxy for stellar age, with an active fraction that is ~ten times higher in the youngest versus oldest post-starburst galaxies. Since a similar trend is seen between age and molecular gas fractions, we argue that, like in local galaxies, the age trend reflects a peak in available fueling rather than feedback from the central black hole on the surrounding galaxy.