Hi Nicole,
Pop ups and alias' for feature layers can be modified and stored in two places. It can be stored with the item or stored with the feature layer or stored in the web map properties. In most cases it shouldn't matter where your pop ups and symbology are stored, unless you want to re-use these settings in other web maps.
Pop up information and alias' can be stored in the web map or with the layer item. If you modify the pop-up or alias' and save the changes with a layer, select the option to save layer in map viewer or the visualization tab.
You can see that the pop up info is stored for this feature layer with the item when you look at the item JSON:
http://www.arcgis.com/home/item.html?id=71e873caaf6c4c2984781018d92a5636
http://www.arcgis.com/sharing/rest/content/items/71e873caaf6c4c2984781018d92a5636/data?f=pjson
In this web map, I changed the alias's to all caps and stored the changes just with the web map, even though I am using the same item as the example above:
http://arcg.is/1W4uDj
http://www.arcgis.com/sharing/rest/content/items/a1738ab64eba47549bff9867d6a8a6a6/data?f=pjson
If you want to update a hosted Feature Services alias' at the server level, you will need to update the service definition through the update definition REST API call.
Feature Layer:
https://services5.arcgis.com/cuQhNeNcUrgLmYGD/arcgis/rest/services/AnimalTelemetry2005/FeatureServer...
If you are making maps in ArcGIS Online, I would recommend saving symbology changes and pop ups to the layer if you want them to be perpetuated through the organization or with the web map if they are a one time configuration. YOu really only need to update at the feature service if you are adding the feature service directly to an application without referencing the itemID (which ArcGIS Online does automatically for you).
-Kelly