Sticky at top of lists node at certain time
Looks like Rules Scheduler (part of the Rules module) is what you're looking for. "Publish content exactly 24 hours after it was created" contains step by step instructions about how to do so (using the Rules Scheduler).

Here are some video tutorials about a few variations of your question (some are pretty close to your question), which are all resolved with Rules Scheduler:


Schedule front page publishing.
Publish a newly created Drupal node on a future date.
Send a follow-up email one week after a user registers.


Ps: There is also the rules-scheduler tag, specific to the Rules Scheduler.