Yes, un-plug the modem from AC power, wait for 10-30 secs then re-plug it into AC. There's also a reboot option available if you login to the modem as admin, but the unplug/re-plug routine works just as well.
The response you received from Centurylink is a canned response. I received the same response. Their first step is just to verify that your credentials (login/password) are correct in their security server. I assume yours are since to are able to view your mail via browser access to webmail.centurylink.net.
One more bit of info may help me: is your Outlook access to your Centurylink mail intermittent, or is it none at all? I've been assuming it was intermittent as mine was.