Since you used the "one-click" installer, you can upgrade to Joomla! 1.0.13 very easily using that same installer.
Just go to the "one-click" section of the panel and select "upgrade" for the Joomla! installation involved; the installer will do the rest (including placing your "old" installation into a backup folder "automagically").
You will not need to do any database upgrading, as there are no changes to the database, and you "upgraded" site should work just fine. You may have to "copy over" your custom images and template changes, but you may not (depending upon whether the "one-click" overwites anything in those directories).
If you do, it is easy to do via FTP or via the shell - you just copy the contents of the appropriate template or image folder from your backup (or the backup DH creates for you) in to the "new" Joomla! directory structure at the appropriate place). In short, you do not have to "start from scratch".
There is a "revert" mechanism provided by the "one-click" robot so that, if something "borks", you can easily put things back the way they were (the email you will receive informing you of the successful upgrade detail how to do that).
Hopefully, I've answered all your questions; it should work a lot "smoother" than your local system upgrade went! .