You can put in a A record in your DNS to point to the other server... however, the other server MUST setup the subdomain in apache. More then that, i would also have the other server duplicate your DNS records incase mail and other things are sent out of the subdomain.
In your WHM under "DNS Functions" click "Edit a DNS Zone" add the subdomain they want such as "sub", make sure the select box is "A" now add the IP of the account on the other server in the text box right after the select box.
Now you have to make sure the account is setup on the other server under the same subdomain and domain with the exact IP you just added into your own WHM.
It's possible. You may do it in many ways the most simple way is to tell him to add a .htaccess file to redirect the subdomain to the desired place.
Redirect /olddirectory http://yoursite.com/newdirectory/
You may also give a redirection in the httpd.conf file. But it would be better to use the .htaccess by the user since it would not mess the httpd.conf file.