Login to your old cpanel account and go to the file manager.
You have to go the the root folder of your old server. The folder before public_html. After going to the root folder you have to work with mainly 3 folders namely mail, etc and .cpanel.

Go to the mail folder and you will find a folder namely yourdomain.com. Zip and download this folder.
Now go the to etc folder, her you will also find a folder namely yourdomain.com, zip and download this folder.
Now from the .cpanel folder downlod two files namely email_accounts.json and email_accounts_count.

Now upload this file to the new server to the same location, unzip it and you are done.