If you provide information concerning your setup ... remotely hosted, shared system (also provide hosting provider name), etc. then answers could be more specific or at least find, for you, the FAQ for customers on those hosted systems. Otherwise, the responses are educated guesses which still might be in-correct.
In Site Admin Menu -> Server -> PHP Info you will see the settings related to time outs.
Doesn't sound like you are on a dedicated server, but rather, a shared system? Thus, you'll have to consult with provider docs/helpdesk on how to increase time limits and other settings (like size of file to upload, etc.).
FTP ... depends upon IF server is set up to accept FTP connections. If so, you'll need a login/password (those aren't related to Moodle) and then you'll have to discover the path to your moodledata directory ... where the repository directory resides. Again, on a shared system, you'll have to consult with provider for specifics.
Now I might be getting confused here ... you are trying to restore a backup made on the SAME server, is that right? Should NOT have to download to restore a course. In any course, click restore and see if you can find the backup (private files?). If you can see the backup, you should be able to restore.
Two Moodle instances on same server and you are trying to restore a course from one instance to another? Is that right? IF on same server and saving backups to a designated area (not inside the moodle file system), then whatever you have to browse files and will allow you to copy files.
'spirit of sharing', Ken