Since you will be "Upgrading Apache" it is clear to me that you have your own in-house server.
Assuming the webserver is only a webserver you could grab any old machine and make a temporary server with the same IP address. Setup the site(s) so that any traffic to the temp. server will get the Scheduled Maintenance page.

Maybe use a 307 Temporary Redirect to get to the Maintenance page.

I'd then shutdown the server neading upgrading and turn on the temp server. Visitors and spiders would then get a friendly message that the site is down for maintenance. Since it would be a 307 temporary redirect, the spiders would continue to look for the proper urls in the future.

The problem I see with involving the DNS and making changes is the update delay. Many visitors pre-update will get an error message and so will the spiders. The temp server and 307TR method avoids this.