The below assumes that you know what you're doing
1. Can I FTP the files?: Yes
2. Do I need to download/save any files I made changes or modifications to and reupload them afterwards?: Yes. You should back up your whole site for that matter. You generally shouldn't just upload the old files. You need to compare the old against the new and add any new code that needs to be there.
3. Also, I would assume I would need to edit the newer of those files and paste in the changes I previously made.. as I would guess some of the files I modified have been updated with the new X-Cart version? - Aye... Addressed in Question 2.
You should really do this all locally on your computer first - setup your computer as a web server and get to work. When all works then you can move your local site to your live server.
The tools I use (Mac OsX)
Cyberduck
Text Wrangler - Text Editor and File Comparing
Terminal (For SSH)
24" Samsung Monitor
Caffeine
Fickle Ego
Etc...
Best,
B
P.S. Carp is right. Get SSH access and google the commands you need to know when you need to know them (like how to change file permissions recursively and such). You'll be glad you did and you'll add a badge to your geek jacket. If your host won't give you ssh, look elsewhere.