System.RestContext is not the same as System.RestRequest.
To be able to successfully test your @RestResource annotated method, you need to feed Salesforce the appropriate data (that is, a System.RestContext).
In effect, there's not much different that you need to do here, as System.RestContext can be though of as a wrapper containing both a System....
As a part of the call to the external system you need to send it the id or some other unique external id field.
The external system can then make a call to either Rest or SOAP Salesforce API or an Apex method exposed as a Rest resource or Webservice to make the update using the unique field sent to it in the first call.