Is it possible instead of downloading a file to my computer and then uploading it to my ftp server, to download it off a website straight onto my ftp server? I've tried googling, but no result.
You would have to invoke this via CLI (command line interface) to achieve it. I don't know of any tools that allow it without using the CLI (but I'd like to stand corrected).
Options:
Direct FTP using DOS (http://www.ics.uci.edu/~cjensen/info/DosFtp.htm)
WGET - Only if you SSH/login to your server in question - which will download the file directly to the server you're on, but you need to be on the server first:
wget ftp://ftp.gnu.org/pub/gnu/wget/wget-latest.tar.gz
I've lost count of the number of times I've done this in the past and skipped browsers...