I understand that the Visibility Range can be set for layers, but I do not see any option to set the range for only labels. Is there a way to have layers visible at a different scale than the entire layer?
One way is to set the visibility scale range of Labels in ArcGIS Desktop.
Else, convert layer to Geodatabase Annotation (in ArcGIS Desktop), and you can set the visible range of the annotation layer in Portal.
The caveat is that the labels only render when the layer itself does, so you can define a label range that is more restrictive than the layer (features show at City scale, labels show at Streets), but you cannot create a label range that is less restrictive.