I can't tell. I can't log in to your control panel. Only you can. Many service hosts do provide FTP access via their control panel, though. I've never used such a service, though. Using a local client is just much easier to me (but then again I think that using ftp via the command line is the easiest way to do it...)
You'd need to be very specific about the file you wanted and make sure they understood. But we don't even know they have a backup yet, so the point might be moot.
Why don't you give them a ring? They're usually pretty helpful, I find. I know lots of people complain about the support they get, but I've always found them helpful. Although I've no experience of this particular company.
BTW, I looked at your earlier post again. Yes, the apostrophe would have done it if you didn't escape it. If you don't understand PHP you should leave functions.php well alone. In fact you should leave all the PHP files well alone unless you're changing them in a child theme and under expert tutelage.