When you create your web map, if you add the layer from web and put in the full rest URL to the layer when you author the webmap, ex..http:// sampleserver1.arcgisonline.com/ArcGIS/rest/services/TaxParcel/AssessorsLiveLayers/MapServer/1http://sampleserver1.arcgisonline.com/ArcGIS/rest/services/TaxParcel/AssessorsLiveLayers/MapServer/1
Configure the Popup, save the map. Opening the map in Explorer on the Iphone may show the links to the attachments. (it worked in my test)
I reproduce what you were seeing with attachments or rather not seeing when I added the service to the map by "Searching for layer" and entering my server, and adding the layer from the list. When I entered the full url to the layer as pictured above, the attachements were accessible in Explorer on the Iphone.
Note the the URL above does not have any attachments it is just an example of what I mean by full URL to the layer.
For your map you can get the full URL by clicking description on the menu.
From the Service rest endpoint click on the layer, and copy the full URL from the browser address.
Mark