How to test drive WordPress on VPS with LEMP stack before going live?

WordPress doesn’t care much about IP address, it is mostly sensitive to domain name it is accessed by.

In your case you can access the test site if you make your local machine resolve domain to its IP (ignoring “actual” IP DNS would resolve it to).

You can accomplish that with hosts file or software that emulates DNS locally (if you need to do that a lot).

While this is perfectly workable I think it is more common to simply stage WP site on dedicated separate domain name. For example that makes easier to give someone link to the site to take a look.