Alexander became arabarches of Alexandria, i.e. chief customs officer. He was therefore accustomed to dealing with large quantities of money, and it is no coincidence that he was chosen as one of the administrators of the possessions of Antonia Minor, the wife of the Roman prince Drusus. He used his profits for several purposes, such as the decoration of the Temple at Jerusalem, and a payment to the Jewish prince Herod Agrippa, who was broke but desperately needed to go to Italy (35/36).
During the reign of Antonia's grandson, the emperor Caligula (37-31), Alexander was in Rome, where he was taken prisoner. We do not know why he went to Italy, nor the reason for his arrest. However, in 38, a pogrom had taken place in Alexandria (go here for the story), and it is possible that Alexander was a member of the embassy that was organized by his brother Philo. As for the reason of the arrest: the fact that he was rich and Jewish may have been important, because Caligula needed money and was quite hostile towards the Jews.
When Caligula was murdered, Antonia's son Claudius became emperor, and Alexander was released. It is possible that he was made a Roman knight at that stage; his sons certainly had that status. Claudius also made Agrippa king of Judaea (41). The new king repaid Alexander for the money he had once received: he married his daughter to Alexander's younger son Marcus. It is likely that Alexander saw that his older son, also called, Alexander, became governor of Judaea (46-48), and it is also possible that he was present when he was governor of Egypt (69).
Tiberius Julius Alexander is a fascinating person: he was some sort of go-between between the Jewish, Roman, Greek, and perhaps Egyptian cultures. In his age, this was still possible; one could be a Jew and could be integrated in the larger Mediterranean world. However, after the pogrom during the reign of Caligula, things started to change. Anti-Semitism became more common, and Judaism started to look inside instead of outside. It is no coincidence that Alexander's son Alexander choose to become a Roman.