How do I setup Apple Mail to automatically move old email to the Archive folder?
"I have a me.com email address." queued me in that you are probably on IMAP. Simply put IMAP syncs your mail across all your devices, instead of having to delete a particular email three different places.

So, to do what you desire, you will have to create a rule on your Mac:


Open Mail Settings and go to "Rules".
Click "Add Rule" and type in your description.
Create the Logic:

Date Received :: Is Greater Than :: [Arbitrary Number Chosen by You]

Under perform the following actions:

Move Message :: to mailbox :: "Archive" -> "iCloud"

Click "OK", then "Apply".
The changes should be reflected on your iPad and iPhone counter-parts after refreshing inboxes there.