I spent about a day and a half in Jerusalem’s Old City on my trip to Israel in November 2019. That appears to be enough to see all of the major points of interest, both in terms of the inscribed WHS and the tentative extension; although in a city with such wealth of historical context, you will feel like you rushed through some of the sights no matter how much time you give yourself.
The very brief Jerusalem WH inscription is all about its significance to the three main monotheistic religions, and being an adherent of one is likely to be an important ingredient for a true spiritual connection to the city. Religious people will surely be awed by being so near the sacred sights, much more so than a secular visitor will ever be.
That secular visitor, nonetheless, will at a minimum find the trio of sights listed on the inscription rather impressive. The Church of Holy Sepulcre is tremendous, the Wailing Wall is sober and full of poignancy, and the Dome of the Rock is simply mindblowingly beautiful. Getting to the Temple Mount requires an extra effort; non-Muslims can enter only as tourists and only at limited times during the day; they are also prohibited entry to the interiors of the buildings in the compound. The visiting hours limitation means that a wait in line of upward of 45 minutes is practically unavoidable. And entirely worth it - the edifice is just stupendous when seen up close.
Beyond those three main sights are many other churches and temples, the unmissable Cardo ruins, the shuk in the Muslim quarter, the city gates, and the car-free narrow streets of its principal four quarters (which happen to be not too distinct visually unless you notice the differences in the appearance of the majority of inhabitants). You can also take a walk on the ramparts (not recommended on hot days) or go for a guided tour of the Western Wall Tunnels, which will provide additional insight into the history of the city. There are several museums of different sizes within the city walls, such as the Tower of David Museum or the Herodian Quarter Wohl Archaeological Museum, to name just a couple that can yet expand your exposure to the history of Jerusalem. If you budget time for these - and other - museums, you may need more than a couple of days in Jerusalem.