Accesing incoming email with PHP in a shared host

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: 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.