There is no option available to publish only part of a Google Site.
You can make edits and not publish the site until you want the edits to go live, but when you publish all edits you have made go live. You can't select which edits are published.
One possible method is to put the edits in a page that is hidden from navigation and unhide it when you want it to go live. Pages hidden from navigation will not be hidden from site search.
You can use a separate unpublished site for working on pages and then use copy and paste to bring the content into the main site, or you can publish the site and link to the pages from the main site.
Hide a Page
When you hide a page, it won’t show up in the menu, but people can still get to it with a link.
On a computer, open a site in new Google Sites.
At the right, click Pages.
On the page you want to hide, click More Hide from navigation.
To publish your changes, at the top right, click Publish.
You can’t hide the home page, but when you hide any other page:
Subpages are also hidden.
Other sites and pages can link to the page.