You are correct, even on normal Ubuntu Linux servers the default wwwroot for Apache is /var/www/.
As mentioned it is designed to run in its own directory. So, in your circumstance you would just drop it in the 'public_html" inside it's own directory. Eg /path/to/public_html/restore/
Then all you need to do hit the restore directory from your browser and follow the steps to restore your blog to 'public_html'.
The reason I opted for this approach is because leaving any of the restore files on your server afterwards is a security risk. Ergo deleting an entire folder afterwards is easier then locating all the individual files.
Hope this helps.