The priority that can be set by Eudora (and other email clients like Outlook, etc...) is simply a line in the email header that some mail clients may use to display the message differently (like having an ! next to it). In general, it has nothing to do with how mailservers process/deliver the email. There is no standard way of marking an SMTP email for any sort of prefered or high priority delivery.

Your best bet is to dig into the mailserver logs and see why the emails are taking so long to be delivered. Some mail server tuning/tweaking may be in order.