Option Explicit
Function SaveWebFile(ByVal vWebFile As String, ByVal vLocalFile As String) As Boolean
    Dim oXMLHTTP As Object, i As Long, vFF As Long, oResp() As Byte

     'You can also set a ref. to Microsoft XML, and Dim oXMLHTTP as MSXML2.XMLHTTP
    Set oXMLHTTP = CreateObject("MSXML2.XMLHTTP")
    oXMLHTTP.Open "GET", vWebFile, False 'Open socket to get the website
    oXMLHTTP.Send 'send request

     'Wait for request to finish
    Do While oXMLHTTP.readyState <> 4
        DoEvents
    Loop

    oResp = oXMLHTTP.responseBody 'Returns the results as a byte array

     'Create local file and save results to it
    vFF = FreeFile
    If Dir(vLocalFile) <> "" Then Kill vLocalFile
    Open vLocalFile For Binary As #vFF
    Put #vFF, , oResp
    Close #vFF

     'Clear memory
    Set oXMLHTTP = Nothing
End Function

Sub TestingTheCode()
     'This will save the Google logo to your hard drive, insert it into the
     ' active spreadsheet, then delete the local file
    SaveWebFile "http://www.google.com/intl/en/images/logo.gif", "C:GoogleLogo.gif"
    ActiveSheet.Pictures.Insert "C:GoogleLogo.gif"
    Kill "C:GoogleLogo.gif"
End Sub

For i = 1 to lastRow
    cellAddress = Replace(Range("A" & i).Hyperlinks(1).Address, "mailto:", "")
    'Something to get the file name from the whole file name here
    SaveWebFile cellAddress, destinationFolder & filename
Next