Pointing a new domain at a site I built in directory outside of root

I built a site for a friend on my server at InMotion. My original plan was to move it to Network Solutions where they purchased the domain. However, they haven't purchased hosting yet and when I asked them to, they asked if I can host it for them. I am on the Power Plan and see that I can add up to 6 websites to my plan. I currently only have knssandbox.com pointed to the root. How would I go about pointing a new domain at knssandbox.com/newebsite and ensure that 404 pages don't resolve at knssandbox.com? Do I need to start over and set up the domain within my account first and then re-install the site or can I simply point the domain to my server, set up the domain in my account to point to that directory and then perhaps run a search and replace script on the directory to ensure all links are no longer using knssandbox.com?

Thanks,
Scott