Is there any way process incoming email using PHP, or Bash, with Cron in a shared server?
Yes, to do it you need to set a filter in the panel to forward email to a shell account.
Go here: https://panel.dreamhost.com/index.cgi?tree=mail.filters& to create the filter. Once of the "Do this'" options is to forward to a shell account. The dropdown in this section should list all shared hosting shell accounts that you have created.