If you have Outlook or any other desktop email client, just set up a new account (in the desktop client) with the POP3 information, and it will download (most POP3's only support syncing the main Inbox folder, if I'm not mistaken) the entire mailbox to your computer.
Then, no matter what happens to the online account, you'll still have all the mails from the account...including offline access.
I recommend that you use POP3, because if you delete something on the online mailbox after it has downloaded to the client, the client won't synchronize that change, so this means that after the account has expired, you'll still have your mails.
IMAP, on the other hand, you can syncronize more folders, but the changes are reflected on both the server and the computer, thus you may lose your mails from the online email account after it expires...