You have one update action for a resource. As a rough rule of thumb a table in your database will be a resource. So in your example there is one controller for the Customer with one update action and fields are optional as far as possible.
There can be exceptions to this (for example if you update the password you expect password & password_confirmation), but this holds true for most cases.
It is more likely that the exceptions are controllers that have limited actions for some resources, like a SessionsController that can only be created and destroyed. It is very unlikely that you would want to have actions for single fields.
In your example one thing could be handled as a resource of its own and that is the address. If you would have a table for addresses, especially if there would be more than one address per customer (billing, delivery...) this could be handled as a nested resource. But again one update action for the whole address resource.