No, it is not possible to specify the order of pop-ups in a web map in ArcGIS Online if there are multiple features in the same location.
When creating a map in ArcGIS Online, there is no capability to specify the display order of the pop-ups because this brings complexity to the user interface. The order the pop-ups are displayed is based on the fastest response returned by the server when a query request is made; in this instance, the selection of the location to display the pop-ups for multiple features. Usually, the server responds fastest for pop-ups of features with the shortest distance to the click point. This is by design in ArcGIS Online for optimization purposes.