First step to take is to reduce the TTL of all of your DNS settings for the domain.
Using the VPS database is not a bad idea, however you would have to test this new WP instance actually works.
To ensure your wordpress is connecting to the remote database, use the IP of the VPS and also make sure the SQL user can be accessed from outside the machine (firewall and SQL setting).
If you are worried about testing, create a DNS record for a subdomain that points to the VPS and change the database settings of that database (not connected to live site) to work with the development URL.
Seeing you have setup a complete new webserver without testing it actually works, I advice you to first test the server by uploading installing a clone of the live site and changing it to work on a temporary URL (as said before).
Another thing that might be important seeing the live website is a webshop is to check whether your licenses will also be working on the new IP.