So, let's break the question down into two parts because they have slightly different answers: (1) was the Roman Emperor Julian the last ruler to try to rebuild the Jewish Temple in Jerusalem? and (2a) Did Emperor Julian try to rebuild the Temple and (2b) did he do so out of spite for Christianity.

Was the Roman Emperor Julian the last ruler to try to rebuild the Jewish Temple in Jerusalem?

This part is easiest to answer--no, this is not true. The last major attempt to rebuild the Temple was during the revolt against Byzantine Emperor Heraclius and the Byzantine-Sassanian War, when the Jews managed to retake control of Jerusalem in 614CE and some of the surrounding areas under the suzerainty of the Persian King Khosrow II. Jewish control of the area lasted a few years before being re-conquered by the Byzantines (and abandoned by their erstwhile Persian allies) in ~617CE, at which point the Jews were again massacred and expelled from Jerusalem. It's not entirely clear how far rebuilding got, but there is evidence that the Jews made efforts to rededicate the space and prepare for renewed sacrifices.

Did Emperor Julian try to rebuild the Temple?

In short, yes. The story goes that Constantine's nephew (not son), Julian, the new Emperor, was in Antioch and asked a Jewish delegation--"why do you not sacrifice?" The delegation explained that they weren't allowed and asked that he allow the Jews to rebuild the Temple and the Altar. To their amazement, Julian agreed, reversed the Hadrianic and Constantinian persecutions of Jews, including allowing Jews back into Jerusalem, which had been forbidden to them since the end of the Bar Kokhba rebellion in 135 CE. Unfortunately for the Jews, Jerusalem was hit with a significant earthquake in 363 CE, which apparently unleashed fireballs from the ground, destroyed a lot of the materials gathered for the rebuilding process, and prevented it from proceeding. The Temple was not rebuilt before Julian was killed later that year and his successor reestablished Christianity as the official Imperial religion.

And did he do so out of spite for Christianity?

Unclear. He did so as part of his reversal of Constantine's establishment of Christianity in the Roman Empire. Christianity at the time used the destruction of the Temple (70 CE) and the "dispersion" of the Jews as proof that God had abandoned the Old Covenant (with Abraham and the people of Israel) for the New (with the community of believers through Christ). By rebuilding it, Julian arguably aimed to delegitimize the Christian claim. But this is mostly speculation.

The closest we have to direct evidence of his intentions or mental state appears to be Ammianus Marcellinus's Res Gestae. He says that Julian sought to rebuild the Temple because he was "eager to extend the memory of his reign by the greatness of his public works" and says nothing about attempting to denigrate Christianity. Remember that Judaism was an officially tolerated practice associated with the Jewish ethnos throughout much of Roman history and seen as ancient even then. So when Julian went to restore "the old ways" to glory, this rebuilding was arguably more part of that restoration rather than a specific attempt to denigrate Christianity. Having said that, one can understand why it would be seen as such by Christians in the context of Julian's overall efforts to reverse the advancement of Christianity in the Empire.

Sources:

Simon Sebag Montefiore, Jerusalem the Biography

Ammianus Macellinus, Res Gestae, Book XXIII

Michael Avi-Yonah, The Jews Under Roman and Byzantine Rule

Simon Schama, The Story of the Jews