Mapping to Domain

Your site must be published to a URL before it can be mapped to a domain.

From Google at 

You can use a custom domain for a site published on new Google Sites

Set up a Custom Domain

Note: You can connect up to five custom domains to one site. It may take up to 48 hours for changes to your domain to become visible on the Internet.

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Custom URL

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See Custom URL instructions for GoDaddy 

For G Suites accounts read more here and see Steegle's instructions here .

For domains registered with Google Domains click here.  Note that when a domain is registered at Google Domains, domain ownership proof is automatic. See also Using Workspace account.

If you want a domain link that goes to a particular page in your site, use forwarding instead of mapping. For instance 

I did this by setting up forwarding in my domain panel to forward the subdomain 'rate' to a page in the site. See the image below to see how I did it in Godaddy.

New Google sites are automatically secure (https) at the native Google URL and at the mapped domain, so it is not necessary to purchase a SSL certificate. If necessary,  use this procedure to refresh the certificate: Remove the custom URL from your site, wait about 15 minutes then add the custom URL again and that will generate a new SSL certificate. 

I  don't see custom domains option

There are 2 reasons that you cannot see the custom domain option in Google Sites: