No. (I think English is strange here; why "the month of July"? Is it because the language retains some archaic notion where "July" is not the name of the month, but another entity (Ave, Caesar!) to whom the month is related?)
Saying "מיץ מהשבוע שעבר" looks to me actually more correct than "משבוע שעבר", and it's what I'd expect in a newspaper. And it sounds OK in speaking, too, but משבוע שעבר is surely more common there.