Currently, you can use the Get Content REST API endpoint to view the status of a page that's been archived via the Web UI, and can see the status is archived, but there is no way to set that status via the REST API.

The Update Content endpoint REST API endpoint allows the status of a page to be set to three values: current, trashed, historical or draft, but not archived. If you attempt to set the status as archived, the REST API returns a 200 response that infers the page status was updated, but it actually isn't, and the page retains the existing status.