This is actually a pretty fascinating time in the history of what I like to call "that big important hill in Jerusalem." The site of the Al Aqsa Mosque and compound today is also call Haram esh-sharif (the Noble Sanctuary), Har Ha Báyit (the Mount of the House) in Hebrew, and commonly the Temple Mount in English. Naturally, all of those names are frowned on by some religious faction. So I'll use Temple Mount, or just the Mount, mostly because it's the easiest to type and very mildly less contentious than the Hebrew and Arabic options.

To start, it depends on what point in Muhammad's lifetime we're talking about. We don't know much about the Mount under Byzantine and Roman rule after they destroyed the Second Temple in 70 CE. Jerusalem wasn't rebuilt by the Roman until 60 years after the First Jewish Revolt, and they built a temple to Jupiter on the Temple Mount, which was sited as a cause of the Third Jewish Revolt in 130.

As Christianity spread through the Roman Empire, Jerusalem became almost entirely Christianized, and after Constantine I started promoting Christianity, the temple of Jupiter and accompanying statues were torn down. As you mentioned, Julian the Apostate tried to encourage rebuilding a Jewish Temple, but nothing came of it, possibly because of an Earthquake. After that point, we don't really know what was on the Mount. A Byzantine Mosaic turned during excavations in the 1930s, but we don't know exactly what it was. A church seems likely, but isn't mentioned in any surviving documents. Regardless, whatever structure housed that Mosaic, was likely there in Muhammad's early life.

However, in 610 CE, the same year that Muhammad started prophesying, the Sassanid Empire defeated the Byzantines and captured Jerusalem. They were aided by an army of Jews, who had revolted against Christian Byzantine rule to try and have more religious tolerance under Sassanid Rule. They actually got their wish. The Jews were put in charge of Jerusalem for the first time in over 500 years, and ran a small client kingdom for the Sassanids for about 5 years. In that time they apparently demolished whatever was left standing on the Temple Mount and started working on a third Temple.

Unfortunately for the Jews, in a bid to try and strengthen their control in the region, the Sassanids let the local Christians take control of Jerusalem. They forced the Jews out of the city and tore down the partially completed temple, turning the Mount into a trash dump until the arrival of Caliph Umar and his army in 637, at which point it became the site of a Mosque.

So for a very brief time of Muhammad's life as a prophet, there was indeed a temple on the Mount. However, his journey to the "Farthest Mosque" is traditionally placed in 620 or 621, half a decade after that partially constructed temple was destroyed. So make of that what you will. I don't really feel like it's my place to decide how that factors into a religious story of the Prophet traveling hundreds of mile in one night, guided by an angel. It seems solidly outside my wheelhouse.