How long does an Unlawful Detainer stay a public record, and what can I do so it does not show up when I apply for an aparment?

In order for the UD not to show up in public records you must have prevailed, not simply had the UD dismissed or "fully satisfied."

As a public record, there is no "dropping off" the radar when it comes to judgments that have been entered against you. If "taking care of it" means paying the money judgment, then a satisfaction of judgment must be filed by the other side and that demonstrates to the rest of the world that the judgment is satisfied. Records in unlawful detainers are "sealed" for the first sixty days following their filing. It requires a court order to continue to have the record sealed and many courts will not do that absent "good cause" and the fact that the judgment will hurt your credit and chances for another apartment are not typically considered "good cause." If the record is unsealed, the damage is done as it is now part of the public record. Its, as we lawyers like to say, like un-ringing the bell.