There are lots of ways to handle this.
If you want things to be fast, then you need test and production at two different IP addresses. Here is a “help” post I wrote on DNS, etc.: We Help You Understand Internet Services — Loud Feed TV Design and Development http://bit.ly/dBxSkj
Otherwise, you can just work fast and do the replacement at an off peak time for site visits.
Depending out what you have done the export from the WordPress dashboard may be sufficient. If this works, then you won’t need to worry about doing a MySQL export and import, which is more complicated and described here next.
You will need to know how to export the MySQL and then how to fix the two URL’s in the settings for the “site” and the “WordPress installation files folder” (different if WordPress is not in the root). You can change these using PHP MyAdmin, but just remember to look for both URLs to change. If I remember correctly, they get stored in the “wp_options” table.