Location Addresses in Content Manager are handled as Child Collections.
As per the ServiceAPI Help on Child Collections (/CMServiceAPI/help/ChildCollections),"edit an existing item by including an object containing a valid Uri and all properties that require modification".
So, the question is how to get the URI of an existing Street or Mailing Address on a Location?
Refer to the the "Fetch any Custom Property" Service API example: http://localhost/CMServiceAPI/examples/CustomPropertyTryItOut?trimType=Location
The requested URL sample generated is /CMServiceAPI/Location/1661?properties=ChildAddresses where 1661 is the location URI. Select the Property required, which in this case is 'ChildAddresses'.