You can make the change on your online billing, however your original log-in/username is your eir email address, which will remain as your log-in/username, even after you change your email address for your correspondence from Electric Ireland.
You will receive your bill notification email to your new email address - it will just be necessary to log in with your old log-in/username.
If you would like to have the online billing deactivated completely so that you can set it up again with your new email address as your log-in/username, email email@example.com
Hopefully this'll also help a bit to spread the word about the eircom.net changes.
(Eir hasn't been doing that thoroughly, and the best place I know for anyone wanting the details is an ongoing topic in the Broadband forum).