Will there be an ‘old’ Jerusalem in Israel during the millennium, since there is a New Jerusalem for the saints to live in?
Yes. At least the northern part of Jerusalem will exist in the Millennium. I think it will be moved further north by the earthquake that splits the city on the day of the Lord’s return (Zech. 14:4) and will therefore be just to the south of the new Holy City where the Millennial Temple will be located. Speaking of the “old” Jerusalem, Zechariah said;
The whole land, from Geba to Rimmon, south of Jerusalem, will become like the Arabah. But Jerusalem will be raised up and remain in its place, from the Benjamin Gate to the site of the First Gate, to the Corner Gate, and from the Tower of Hananel to the royal winepresses. It will be inhabited; never again will it be destroyed. Jerusalem will be secure. (Zechariah 14:10-11)