Cush was telling us:
Cush nous racontait que :

Make sure you do not use the selection object.
Set a range to the paragraph you want to insert the floating picture,
then set a range to a paragraph below where the freshly pasted picture is
anchored and add your text there.

Dim appWord As Word.Application
Dim rgePaste As Word.Range

'Not how you did it from Excel...
'But I tested in Word...
Set appWord = Word.Application
Set rgePaste = appWord.Selection.Range

With rgePaste
   .InsertParagraph
   .Collapse wdCollapseStart
End With

AppActivate "Microsoft Excel"
Sheets("Congress_Charts").Select
ActiveSheet.Shapes("Report1 - Pic01").Select
Selection.Copy

With rgePaste
   .ParagraphFormat.Alignment = 1
   .PasteSpecial , False, wdFloatOverText, False, _
       wdPasteMetafilePicture
   Set rgePaste = .Next.Paragraphs(1).Range
End With
With rgePaste
   .Collapse wdCollapseStart
   .Text = "We need to invest in high schools..."
End With

Also, in Excel you should not need to use AppActivate and Select.
Use a shape object to refer to your picture and use that instead:

Dim wksTarget As Worksheet
Dim shpTarget As Shape

Set wksTarget = ActiveWorkbook.Worksheets(1)
Set shpTarget = wksTarget.Shapes(1)
shpTarget.Copy

It is much more efficient and much less error-prone to use objects.
With object you never need to use the selection object and actually have the
code go from one window to the other...