Für alle die das gleiche Problem haben untenstehend der komplette Code (zusätzlich noch ergänzt um die Ausgabe, wie viele Dateien abgelegt wurden):

Option Explicit
Private Declare Function URLDownloadToFile _
                          Lib "urlmon" Alias "URLDownloadToFileA" _
                              (ByVal pCaller As Long, ByVal szURL As String, ByVal szFileName As String, _
                               ByVal dwReserved As Long, ByVal lpfnCB As Long) As Long

Sub Hyperlink_Downdload()
Dim olMsg As MailItem
    On Error Resume Next
    Set olMsg = ActiveExplorer.Selection.item(1)
    DownloadLinkedFile olMsg
lbl_Exit:
    Exit Sub
End Sub

Sub DownloadLinkedFile(olItem As MailItem)
Dim DateiAnzahl As Long
Dim olEmail As Outlook.MailItem
Dim olInsp As Outlook.Inspector
Dim wdDoc As Object
Dim oRng As Object
Dim oLink As Object
Dim vAddr As Variant
Dim strFName As String
Dim strURL As String
Dim strLocal As String
Const strPath As String = "C:\....\"      'Pfad für die Dateien


    On Error Resume Next
    With olItem
        Set olInsp = .GetInspector
        Set wdDoc = olInsp.WordEditor
        Set oRng = wdDoc.Range

        For Each oLink In oRng.hyperlinks
             If Right(LCase(oLink.Address), 3) = "pdf" And _  'Dateiendung (wenn angepasst wird auf die Anzahl der Zeichen achten)
               Left(LCase(oLink.Address), 5) = "https" Then
                vAddr = Split(oLink.Address, "/")
                strFName = vAddr(UBound(vAddr))
                strURL = oLink.Address
                strLocal = strPath & strFName     
                Call URLDownloadToFile(0, strURL, strLocal, 0, 0)
                DateiAnzahl = DateiAnzahl + 1
              End If
            
        Next oLink
        
        MsgBox DateiAnzahl & " Dateien abgelegt unter " & strPath
    End With
    

lbl_Exit:
    Set olInsp = Nothing
    Set oRng = Nothing
    Set oLink = Nothing
    Exit Sub
End Sub