I heard that if I purchase Navigo Week pass on Friday, it is issued for the next week.
Is that right?
I want to use it on this Saturday and Sunday though there is only two days left.
Yes, if you buy Navigo week pass on Friday, it's valid starting Monday and cannot be used Saturday / Sunday.
If you want the same unlimited use coverage through zones 1-5, put Navigo Jour (day pass) onto your Navigo card. If you don't already have a Navigo card then buy a Navigo Easy card (2€ vs. 5€) and put two Navigo Jour day passes on it for Saturday and Sunday. If you're using Navigo Découverte it can only hold a single day pass on it at any time. (Navigo Easy allows 2 day passes at a time.)