I had a similar problem, adding a map service (published by myself in a regional coordinate system, Belge_Lambert_72) on top of one of the ArcGIS Online basemaps in the webmap application on ArcGIS.com. The map service data had a fixed shift, due to missing datum transformation. I knew the issue in ArcMap, where I have to select a specific datum transformation, in order to make the layers match. How to make the layers match in the webmap application?
The only think I had to do is add the datum transformation of my choice with ArcMap in the MXD. Even when you don't have to select a datum transformation in ArcMap, because all layers are in the same, local coordinate system, you can still explicitly specify datum transformations the usual way. After saving this to my MXD and restarting my map service, the layers matched and the shift was gone.
Kristof Vydt