Did Muslims occupy Israel before the Jews?

Answer 1

No:

The Israelites (early Jews) were in Israel (Canaan) for 220 years, in 1700-1500 BCE. Later, the era from Joshua until the First Destruction (including the Judges and Kings) was 850 years. The Second Temple era was, according to traditional chronology, another 420 years (not 586), which included the Hasmonean dynasty. That's a total of 1490 years.
After the Second Destruction, there were thousands of Jews who remained in Israel (Judea; Palestine) throughout the Talmudic era and beyond (see for example the Talmud, Sanhedrin 17b). They were the majority of Palestine's population well into the fourth century, with records attesting to at least 43 Jewish communities, most of them in the Galilee and Jordan valley. After that, there were still Yeshivas in Israel with at least some thousands of community-members.
In the fifth century, the Jerusalem Talmud was completed in the yeshiva of Tiberias, by the disciples of Rabbi Johanan; and the Christians of Palestine declared Judaism to be a tolerated minority.
In the sixth century CE, Mar Zutra and his descendants served as head of the community and the Rabbinical academy in Tiberias.
In the seventh century, the Palestinian Jews joined the Persians in a battle to take Jerusalem from the local Byzantines, and enjoyed a brief autonomy, which the Byzantines under Heraclius officially recognized in 628. At the time of the Moslem conquest of Palestine in 638, the Jewish population has been put at no less than 300,000; and a period of flourishing began. Caliph Umar encouraged Jews to resettle Jerusalem.
In the eighth century, there were 30 synagogues in Tiberias. A Jew named Abu Issa brought his forces in battle against the Caliph.
In the ninth century, the Jews of Palestine instituted their own Gaon (leading sage) in Tiberias and later in Jerusalem.
In the tenth century, we have the greatest of the Massoretes, Aharon ben Asher and Ben Naphtali, flourishing in Tiberias.
Contemporary with Rashi (11th century), we have a Rabbi Abiathar and others, who lived in Israel (see for example Rashi commentary, Talmud Berakhot 62a), and large Jewish communities in Rafah and Ramle, Hebron, Acre, Caesaria, Jaffa, Ashkelon and Gaza.
In the 12th-13th centuries, the Palestinian Jews were harshly persecuted under the Christian Crusaders, yet many Jews continued to live in all the above-mentioned towns as well as Haifa, with Judah Halevi journeying to Palestine in 1141, Maimonides in 1165, and Nachmanides in 1286. In 1187, Saladdin invited more Jews to settle Palestine. In 1204, a group of Maghreb Jews arrived; and in 1211, 300 Rabbis arrived from France and England. In 1260, Rabbi Yechiel of Paris established a Talmud academy in Acre.
Since that time, the continual presence of Jewish communities in Palestine (Israel) is well-known and needs no reiteration.

Answer 2

Yes. Judaism, Christianity and Islam all have their origins in Abraham. Judaism is the oldest, then Christianity and then Islam.