I am actually the developer of the Zipit Backup Utility. I just happened to find this.
For restoring your site it should be fairly easy. You should have two files in their respective containers.
1 for the database and 1 for the files.
You will need to extract the files archive locally. You will then have the timestamped folder and inside you will see the lib logs and web folder.
Inside the web folder you will see the content folder. That is the folder that contains your site files. You can then use FTP to login to your Cloud Site and navigate to the content folder of your site and copy the contents of your local content folder to it.
Now for the database, you will need to extract the archive file for the database locally. This will give you the sql file for your database. If you still have the database in Cloud Sites created you can login to phpMyAdmin from the Cloud Sites control panel and drop all the tables from the old database. Then you can use the Import option in phpMyAdmin to import the sql file.
One thing to keep in mind is that when you export an sql file you typically end up with some stuff at the top of the sql file for creating the database if it doesn't exist. This can cause issues when importing in Cloud Sites since it is not allowed to create databases from phpMyAdmin in Cloud Sites. To work around that you can simply edit the sql file prior to importing and remove those lines from the sql file.
While we do not support this type of thing officially. I would be happy to assist you with this. If you can give us a call at our support number that you can find in your Cloud Sites control panel and ask for Jereme Hancock I will do my best to help where I can.
Also, fyi, I just release a major update to Zipit that you might want to check out. :)