Keep in mind, you'll have to move WordPress twice in this senario.
Since your Domain is staying the same, it's actually pretty safe to just FTP the files over, import the Database, edit your wp-config.php to have your new DB and username/password info, and then flip DNS to point to us. It should work.
I will note that there's going to be some downtime, and before you transfer your DNS, you may want to lower your Time To Live to speed that up. Technically a DNS move can take up to 72 hours to propagate worldwide, and there's not a whole lot we can do to make that faster