I’m sorry, I gave the paths wrong before. It is not ~/support; I should have written /users/support – each of your email addresses will be a separate shell user. The process would be to make a new User from the control panel:
This creates a shell account, and then you can set up procmail using the ~/.procmailrc file so that mail delivered to that shell user is processed as you wish.
You might have difficulty setting up a shell account named “support,” or the other generic names you described, if those shell users are already taken or are reserved on your shared hosting machine. But you should at least be able to set up a “mydomainsupport” shell user (or something with a similarly unique name), and then you can use email aliases to send support@ to mydomainsupport@.