Great question!  There's a very old plugin that people used for this:
https://wordpress.org/plugins/cron-unsticky-posts/
It uses the wp-cron script to trigger removing the sticky attribute from a post.  However, it hasn't been updated in years and is only guaranteed to be supported up to wordpress version 2.8.  I'm going to ask one of our authors if they have any other plugins they can suggest for this.