If the external server has a LAMP stack (and is fully functional), then it appears the question you are asking is:
a) How to transfer files to the server:
For this, my recommendation is WinSCP to transfer the files. You will use your SSH password/key and username (exactly as you would for PuTTY) and it will log you into the server. You can just drag files into the right folder. The advantage of this is that if you don't have an FTP server running on your server it will still work. (If you do have an FTP server running, this still offers better security, but FTP may be easier to use).
b) Where to put the files:
This is a bit harder to answer. Typically website files are placed in /var/www/html however, some setups place them in the user's home folder (and symlink them elsewhere) or put them in a completely different directory. If you use FTP (and someone else has set it up) you will typically be brought right to the folder you need to put your files in. Otherwise, it is quite dependent on your setup (look in httpd.conf (/etc/httpd/conf/httpd.conf) for the clues as to where Apache expects the files - that is a bit beyond the scope of this question/answer.
c) If you are looking for moving and/or using your SVN repository:
You may be able to mirror the repository, depending on how it has been setup. You might also simply download a working copy of the repository to the correct directory (depending on the setup, you might even be able to use a post-commit hook to automatically update the site when you commit changes to the repository, but the merit of that is debatable).
You probably have databases, to copy them over either create a dump (using mysqldump) and then import into your database; or simpler (if a bit slower) use phpMyAdmin for a graphical interface to your database. (Your remote server might not have phpMyAdmin - depending again on the setup).