What do you want it to say for the week:

28 May - 04 June?

It would be pretty tricky to come up with code the show 04-11 May or 28
May - 04 Jun.

However, if you can live with 04 May - 11 May and 28 May - 04 June that
would be pretty easy.

Enter field brackets CTRL+F9.  Insdie the brackets, type DATE \@ "dd MMM"

Toggle the code ALT+F9

The field shows 04 May

Select the field text and bookmark it "Start"

Type a dash and then bookmark the IP "End"

Use this macro:

Sub AutoOpen()
Dim oDate1 As Date
Dim oDate2 As Date
Dim oRng As Word.Range
Set oRng = ActiveDocument.Bookmarks("End").Range
oDate1 = ActiveDocument.Bookmarks("Start").Range.Text
oDate2 = DateAdd("d", 7, oDate1)
oRng.Text = Format(oDate2, "dd - MMM")
ActiveDocument.Bookmarks.Add Name:="End", Range:=oRng
End Sub

If you create a new document based on your template (not open the template)
you can use createdate fields and calculation fields. This is more complex
than you might imagine, but it can be done. See
http://addbalance.com/word/datefields2.htm for information on the different
kinds of ways to make a date calculation work. It includes links to
utilities to create the fields and an explanation of different macros that
can be used instead of fields. It also has a link to
www.wopr.com/cgi-bin/w3t/showthreaded.pl?Number=249902 which is a document
with various fields already created. Another resource is
http://www.gmayor.com/insert_a_date_other_than_today.htm.