How to Schedule a node "Promoted to front page"?
You can do this using the Rules module in combination with the "Rules Scheduler" (a submodule of it). After you enable these 2 modules, create a Rules Component and a Rule. Here is a blueprint of doing so:


Create a Rules Component which accepts "node" as a parameter, and which has a Rules Action to set the node to "published".
Create a Rule like so:


Rules Event: Before saving a new node.
Rules Actions: (1) set the node to "unpublished" (to be sure it is not "published" when the node is saved) and (2) schedule the Rules Component created in the previous step.



Note: From the comment below this answer (which states "The node will remain Published all the time.") it doesn't seem to apply for the question here, but using a similar approach you could also schedule another Rules Component to mark it unpublished again later on, eg X days after the node gets published. In that case, you use Rules Event "After updating an existing node", combined with a Rules Condition to check if the update changed "unpublished" to "published".