For some reason I thought you might have local access to your server. If you have to establish remote access to the server file system, then FileZilla using SFTP/SSH connection is really the easiest way. You could
install the Teamviewer service on Linux, but I've never tried that before, and it's really overkill if all you need is access to the remote file system.
If I were you, I would try to figure out why FileZilla isn't working for me. Maybe SSH/SFTP is configured to run on a port other than 2222? Maybe you entered the wrong root username and/or password?
Clearly SSH is working on your server. What port number do you enter into Putty? Try entering that same port number into Filezilla.
If SSH is working via Putty, then it seems like your FileZilla connection is somehow not set up correctly.