Is there software that can be used besides FileZilla to work on the website? Because if you use FileZilla, my work cycle will be very long because you have to download a file you want to edit, then upload it. Is there an FTP client application that can be used without having to download the file first?
No, such a software doesn't exist. It's hard to prove a negative, but FTP is a file transfer protocol, so it transfers files from the server to the client.
Instead, you should be looking for editing software on the server. Some hosting control software has this built-in, e.g. cPanel offers an HTML editor.
Editing files without downloading them cannot be done with a locally installed application, though there are tools which download/upload the files automatically in the background. One of those tools is UltraEdit.