Uhhh, what's ... 'whole site backup' mean? If it means what I think it means, I'd say one course extract isn't currently possible. But, care to share what got broke? Don't suppose you could restore the old site on a test box, check out the course of add-ons, etc., get a backup of the troubled course, and try the restore of the course.
'spirit of sharing', Ken
Well, this is interesting. I have read now http://docs.moodle.org/25/en/Site_backup
I had the assumption that you pressed a button and everything got backed up, but it's not quite like that now I look at things.
Multiple course backups are no (practical) use to restore a site to where it was before an upgrade.
Re our problem: one course (that I know of) came through an upgrade with an assignment missing all the annotated content, with PDF annotate plugin. On worst case scenario, I was wondering what to do if we needed the data in the assignment.
In the end, fixing the plugin seemed to lead to restoring the missing assignments. Which is a little learning in itself.