Easiest way to duplicate directory over FTP

What you have is not a unix command line, what you have is an FTP session. FTP is designed primarily to upload and download files, it's not designed for general file management, and it doesn't let you run arbitrary commands on the server. In particular, as far as I know, there is no way to trigger a file copy on the server: all you can do is download the file then upload it under a different name.

Some servers support extensions to the FTP protocol, and it's remotely possible that one of these extensions lets you copy remote files. Try help site or remotehelp to see what extensions the server supports.

If you want a unix command line, you need remote shell access, via rsh (remote shell) or more commonly in the 21st century ssh (secure shell). If this is a web host, check if it provides ssh access. Otherwise, contact the system administrator. But don't be surprised if the answer is no: command line access would be a security breach in some multi-user setups, so there may be a legitimate reason why it's not offered.

Here's how to do it with lFTP:

# lftp ftp_host
user ftp_user ftp_pass
mirror source target (download entire directory tree)
mirror -R source target (reverse mirror; upload entire directory tree)