Database Migration - Can it be this easy?
High level your approach will work but there is an additional step you will need to do along with a gotcha. 

First off the gotcha - you will need to ensure that the instance names are identical on each server (although you can have a different server names). This isn't an issue though if you are just using default instances. 

Once you cut over the databases you can then rename the new server to the old name. Once renamed you will need to rename the server name within SQL Server (again not to be confused with the instance name) using the steps outlined in this article. 

https://msdn.microsoft.com/en-us/library/ms143799.aspx

I would either use log shipping or mirroring for the 5 databases and then manually copy across each instance level object (logins, jobs, linked servers, mail profiles, etc) and goes without saying that the applications should be tested before hand to make sure you are aware of all the instance level components and don't forget anything. 

I have used approaches like this before to migrate clustered instances with numerous databases. This approach has more work up front getting everything set up and ready but it makes the cut over relatively quick and painless for the client applications.