Assuming you are transferring the existing site domain ...
1. Set up a new account with the name of the domain to be transferred.
2. You can can access that web space by ftp and upload the site to there (must be into the public_html directory there).
3. You can accesss the site before the domain is transferred by using the URL 'IP address/~username' and test it to make sure it's all working.
4. It is possible that all functions won't work properly on the site (I have had some links not working) using this method, but most will, and you can see that everything is there ok. It should all work properly once the domain transfers.
5. Change the domain's DNS to your new nameserver at Hosting Provider.
6. Allow 24 - 48 hours for the domain to propagate to the new DNS and off you go.
If you do that all properly the website should not even go off-line. The only thing you might need to be careful with is if the site has a forum or something else that visitors are adding data to regularly. In this case any data entered between the time you download the original site (for uploading to the new space) and the time the site propagates to the new space will have fallen between the two and be lost. If this applies you should maybe think about closing the forum from the time you downloaded it for transfer until the site has propagated in it's new home.