You can transfer messages from your old webmail system to the new one by simply forwarding them to your new emailaddress.
Although the old webmail service has been disabled, you are still currently able to send emails. However, all 'incoming' emails will be retrieved via the new webmail system.
Therefore, if you wish to keep an important email which currently resides in your old webmail inbox just forward it to your current email address, e.g. email@example.com to firstname.lastname@example.org
The email will then be sent to your new webmail account.