I answered false although I think there are more choices than "it was and always will be Arab" and "was Jewish and will become so again".
Historically, it was only in Jewish hands either by control or major occupancy for about 1,000-1,500 years. Joshua is described as having fought the king of Jerusalem Adonizedek and winning, but some parts of Jerusalem still belonged to the Jebusites until David's time about 450 years later. It remained in Jewish control for the next ~ 450 years until the Babylonians conquered the Judean kingdom. 70 years later, although not necessarily in complete control of the region, the majority of the population was Jewish. About 400 years later the Romans came and took control of the area and enacted measures that caused the Jewish population to mostly scatter over the following few hundred years. Since then it's been variously under Christian or Muslim control.
So from a strictly historical perspective, it has at times been part of the Egyptian region, under various Canaanite rulers, Jewish control, Babylonian, Roman, Christian, European, Persian, Muslim and Turkish rule. I'm sure there must be more. It's false that it was always an Arab city and it's also misleading to say that it was a Jewish city without a nod to the plethora of other regions/religions that it was part of.
From a religious perspective, religious Jews believe that the land was given to us by G-d and there are many commandments that can only be fulfilled in the country and city. We've similarly been mourning the destruction of the city praying for the our return to the country and the city for longer than Islam has existed. It's somewhat more of a complicated issue because the Talmud teaches that we are forbidden to conquer the country, although there are different interpretations to the exact parameters of the prohibition. So we do believe that we will one day return to having ruler-ship of the city and country although that is generally believed (by religious Jews) to be part of the Messianic Age.
At the same time, early Islam saw itself as a continuation or supercession of Judaism by way of Christianity. Early Islamic coinage in the region was found to depict the classically Jewish symbol of the seven branched menorah along with the shahada. As with other religions that descend from Judaism, part of the inherited theology includes the importance of Israel and Jerusalem. So I think it no more wondrous to see Islam claim the city for itself than it is for the Crusaders to fight for it in the name of their own religion.