You would use a rule on the drop-down list that sets the AdminName. It gets the name from the secondary DOM that pulls its data from your SP list. To get the right name, you need to filter the list based on ManagerID = the selected value in the manager field.
You can use the InfoPath UI to do all of this, but pseudo-code-wise it would look something like:
Set field:AdminName = SpList/AdminName[../ManagerID = field:Manager]