I don't really think there is a safe way to make changes to your functions file, and for that matter any other template file. And I personally think that the back end editor are a big evil.
It is unwise to make any changes to a live site, period. Any type of syntax or fatal error will crash your site, and the only way to fix this is to log via ftp and deleting the changes you've made. On a high traffic site, this is disastres.
Always always set yourself up with a local test site. There are many around, and are quick to set up on your pc. I personally use XAMPP, but that will be totally up to you. The advantage of such an offline test site is that you can test code and fix all possible errors before you make this changes to your live site. And if there are errors in the code and it breaks your test site, it can be quickly fixed.
I would also suggest that if you need to make changes to a template, download it to your local installation, make the changes, test it, and if you are happy with your changes, upload it back via ftp. Never use the backend editor for this
Misread your question a bit, only realized that you don't have ftp access when I eddited your question.
So how does this influence what you should do? Well, you still have to set yourself up with a test local site. The problem is, if you've made changes to a template and it works offline, you can't just simply ftp it back to your live site. Here, unfortunately you have to use the back end editor. I would suggest, delete all the content of the template, and then copy and paste the complete offline template to the online template. This will be the easiest way to make sure that an error doesn't slip in anywhere
I would also suggest that you look for a host that support cpanel.