Yes, it is possible to adjust the placement of labels depending on the type of feature used on the web map. Although custom label placement is not currently available, ArcGIS Online does provide alignment options when managing labels for line and point feature types. These options are designed to simplify the positioning of labels and provide an effective, easy-to-use way of aligning labels on the web map.
For lines and polylines, any one of the three following alignment options can be used to display labels: Above the Line, On the Line, or Below the Line. The nine-grid 'tic-tac-toe' pattern is used for the label position of points: Top-left, top-center, top-right, middle-left, center, middle-right, bottom-left, bottom-center, or bottom-right.
Adjusting the position of labels, however, is not supported for polygon features. By design, labels are placed on the centroid of the polygon. Additionally, if there are label-position conflicts between two or more layers of the web map, only the label at the top-most layer displays.
Use ArcGIS Desktop to solve issues with label placement. To do so, open the map in ArcMap or ArcGIS Pro for a better customization of where labels can be placed in relation to the features on the map, and republish the map to ArcGIS Online. Refer to ArcMap: Using the Label Manager for more information.