You should be using a style to apply to the caption rather than doing a local format for bold. Working code might be along the lines of the following
Code:
Sub UpdateTables2()
Dim rng As Range
Dim aTbl As Table
Dim aRow As Row
Dim x As Long
Dim MyTables As Long: MyTables = ActiveDocument.Tables.Count
Dim MyNote As String: MyNote = "Does this table have a table name"
Dim answer As String
Dim aRng As Range
Application.ScreenUpdating = False
For Each aTbl In ActiveDocument.Tables
Set aRng = aTbl.Range
aRng.MoveStart Unit:=wdParagraph, Count:=-1
aRng.Select
ActiveWindow.ScrollIntoView aRng, True
Set aRow = aTbl.Rows.Add(BeforeRow:=aTbl.Rows(1))
aRow.Range.Cells.Merge
Set aRng = aRng.Paragraphs(1).Range
aRng.MoveEnd Unit:=wdCharacter, Count:=-1
aRow.Range.Cells(1).Range.FormattedText = aRng.FormattedText
aRng.Paragraphs(1).Range.Delete
aRow.Borders(wdBorderTop).LineStyle = wdLineStyleNone
aRow.Borders(wdBorderLeft).LineStyle = wdLineStyleNone
aRow.Borders(wdBorderRight).LineStyle = wdLineStyleNone
aRow.Range.Style = "Caption"
Next aTbl
Application.ScreenUpdating = True
End Sub