You'll first need to access the geometry and attributes of the service and store it in a feature set, or save it to disk as a feature class. After a quick search, this can be done using a few tools that are available:
http://epro.maps.arcgis.com/home/item.html?id=16e5cc64178941839eca62837f168ec9
Download ArcGIS Online Feature Service or ArcGIS Server Feature/Map Service
You don't really need to store the feature class on disk. Simply make a request to the query endpoint of the service, and store the results in a feature set. If there are more than 1000 features and the maximum number of records returned by a query is 1000, (which is the default), you'll need to make additional query requests to get all of the data. Once you have the feature set, you can use that as input with your other feature class to run the intersect.
