"What is the quickest way to pack my old Perl installation and move it to the new server? "
Don't. Install whatever version of perl you want elsewhere on the system and leave the vendor perl alone. Otherwise you could end up with an unstable or broken system.
It's trivial to use either perlbrew or the method descried in the README (links here). Once you have perl installed you could use cpans autobundle function to build a list of the installed modules from your existing server, and simply install the bundle it creates on your new server. Before doing so you can edit the file and remove any modules you don't want. See Re: List of Installed Perl Module in Unix Box.