cPanel has a good feature for backup directly from your current cPanel server to another, so you do not need to do manual backup, download, and then reupload again. Here is the way to do it.
First you need to go to backup wizard from your cPanel
After that you choose Backup/Restore feature
For the step 2, choose Full Backup, because partial backup does not have Remote Backup feature
In the last step, you choose Remote FTP Server from list box menu, and then you will be asked to fill your another cPanel FTP data.
Click generate backup, and all you need to do is wait until the process is complete. The time required depends on how big is your file or the speed of your cPanel server, but it obviously faster than do a manual backup. Once the process is complete, you will be notified via email you entered earlier.