We present 24 μm photometry of the intermediate-age open cluster Praesepe. We assemble a catalog of 193 probable cluster members that are detected in optical databases, the Two Micron All Sky Survey (2MASS), and at 24 μm, within an area of ~2.47 deg². Mid-IR excesses indicating debris disks are found for one early-type and for three solar-type stars. Corrections for sampling statistics yield a 24 μm excess fraction (debris disk fraction) of 6.5% ± 4.1% for luminous and 1.9% ± 1.2% for solar-type stars. The incidence of excesses is in agreement with the decay trend of debris disks as a function of age observed for other cluster and field stars. The values also agree with those for older stars, indicating that debris generation in the zones that emit at 24 μm falls to the older 1-10 Gyr field star sample value by roughly 750 Myr. We discuss our results in the context of previous observations of excess fractions for early- and solar-type stars. We show that solar-type stars lose their debris disk 24 μm excesses on a shorter timescale than early-type stars. Simplistic Monte Carlo models suggest that, during the first Gyr of their evolution, up to 15%-30% of solar-type stars might undergo an orbital realignment of giant planets such as the one thought to have led to the Late Heavy Bombardment, if the length of the bombardment episode is similar to the one thought to have happened in our solar system. In the Appendix, we determine the cluster's parameters via bootstrap Monte Carlo isochrone fitting, yielding an age of 757 Myr (±36 Myr at 1σ confidence) and a distance of 179 pc (±2 pc at 1σ confidence), not allowing for systematic errors.