We apply a population-orbit superposition metho1d to 16 galaxies in the Fornax cluster observed with MUSE/VLT in the context of the Fornax3D project. By fitting the luminosity distribution, stellar kinematics, and age and metallicity maps simultaneously, we obtained the internal stellar orbit distribution, as well as the age and metallicity distribution of stars on different orbits for each galaxy. Based on the model, we decompose each galaxy into a dynamically cold disk (orbital circularity λz ≥ 0.8) and a dynamically hot non-disk component (orbital circularity λz < 0.8), and obtain the surface-brightness, age, and metallicity radial profiles of each component. The galaxy infall time into the cluster is strongly correlated with galaxy cold-disk age with older cold disks in ancient infallers. We quantify the infall time tinfall of each galaxy with its cold-disk age using a correlation calibrated with TNG50 cosmological simulations. For galaxies in the Fornax cluster, we found that the luminosity fraction of cold disk in galaxies with tinfall > 8 Gyr are a factor of ∼4 lower than in more recent infallers while controlling for total stellar mass. Nine of the 16 galaxies have spatially extended cold disks, and most of them show positive or zero age gradients; stars in the inner disk are ∼2 − 5 Gyr younger than that in the outer disk, in contrast to the expectation of inside-out growth. Our results indicate that the assembly of cold disks in galaxies is strongly affected by their infall into clusters, by either removal of gas in outer regions or even tidally stripping or heating part of the pre-existing disks. Star formation in outer disks can stop quickly after the galaxy falls into the cluster, while star formation in the inner disks can last for a few Gyrs more, building the positive age gradient measured in cold disks.