If I'm understanding it right, the idea is that the DNS configuration for company-staging.com is identical to that for company.com - same IPs and servers, DNS provider, domain registrar.
Literally the only differences are s/company/company-staging/, all accesses should hit the same server with the same request other than the Host header.
Then you can update the DNS configuration for company-staging.com, and if that doesn't break there's very little scope for the update to company.com to go differently.
Then you can update the DNS configuration for company-staging.com, and if that doesn't break there's very little scope for the update to company.com to go differently.