Personally, I like using Amazon AWS Route 53 and Amazon AWS S3 for this.
S3 lets you host a static "site down" page. Route 53 is their DNS service. You can define a health check (aka, port 80 up ...), and if so if gives out the normal IP address for your system. You then define a failover IP address which is the S3 static web site (sorry down right now, back soon).
This way everything is automatic.
Otherwise a simple fix is on your border switches (assuming they are layer 3) is to create a new layer 3 interface that is a /30. Make the temporary web server the same as the main production web site IP address, and obvisouly the other IP address goes on the switch. Being a directly connected interface it wont do anything normally if it is unplugged. When you are ready to do your changes just plug in the temporary web server and being a directly connected interface it will take precedence over any other routing you have setup.