Use these directions to create a “mirror” http://wiki.dreamhost.com/DNS_-_Viewing_site_before_DNS_change
Instead of creating and using the dreamhosters.com sub-domain discussed in the article you can use your alternative-domain-name.org
As a footnote, if you care about google search ranking’s you will want to add a robots.txt file to tell the google bot to go away for now. After the DNS has changed, you will want to remove or adjust that so the site will be searched and delete the mirror.
Google penalizes duplicate content on sites.