We present new Spectrographic Areal Unit for Research on Optical Nebulae (SAURON) integral-field spectroscopy and Swift Ultraviolet Optical Telescope (UVOT) observations of molecular outflow host galaxy NGC 1266 that indicate NGC 1266 has experienced a rapid cessation of star formation. Both the SAURON maps of stellar population age and the Swift UVOT observations demonstrate the presence of young (<1 Gyr) stellar populations within the central 1 kpc, while existing Combined Array for Research in Millimeter-Wave Astronomy CO(1-0) maps indicate that the sites of current star formation are constrained to only the inner few hundred parsecs of the galaxy. The optical spectrum of NGC 1266 from Moustakas & Kennicutt reveal a characteristic poststarburst (K+A) stellar population, and Davis et al. confirm that ionized gas emission in the system originate from a shock. Galaxies with K+A spectra and shock-like ionized gas line ratios may comprise an important, overlooked segment of the poststarburst population, containing exactly those objects in which the active galactic nucleus (AGN) is actively expelling the star-forming material. While AGN activity is not the likely driver of the poststarburst event that occurred 500 Myr ago, the faint spiral structure seen in the Hubble Space Telescope Wide-field Camera 3 Y-, J- and H-band imaging seems to point to the possibility of gravitational torques being the culprit. If the molecular gas were driven into the center at the same time as the larger scale galaxy disk underwent quenching, the AGN might be able to sustain the presence of molecular gas for greater than or similar to 1 Gyr by cyclically injecting turbulent energy into the dense molecular gas via a radio jet, inhibiting star formation.