It would nicer if the Esri application had a refresh data button, but here is a workaround:
'Copy cells one by one to force the Esri Maps for MS Office map to update
Dim FirstCell As Range
Dim FirstTargetCell As Range
Dim i As Integer
Set FirstCell = Sheets("SourceSheet").Range("BG11")
Set FirstTargetCell = Sheets("GISReadySheet").Range("BG2")
i = 0
Do
FirstTargetCell.Offset(i, 0).Value = FirstCell.Offset(i, 0).Value
i = i + 1
Loop Until FirstCell.Offset(i, 0).Value = ""
End Sub