No, the story of Jesus was written in such a way to appear as though it fulfilled Old Testament prophecies. However, a number of errors make by the Gospel writers (the virgin birth, the 30 pieces of silver, etc), demonstrate that instead of Jesus fulfilling prophecy, they actually combed through the Old Testament looking for things to tie in.
For example, the whole virgin birth story is based on a mistranslation in the Greek Septuagint. It's rather strange that if Jesus existed and were actually God, that he would fulfill a recently made prophetic typo, as opposed to what the actual Hebrew scriptures said.
Consider also this: Matthew 2:23 - "And he came and dwelt in a city called Nazareth: that it might be fulfilled which was spoken by the prophets, He shall be called a Nazarene."
Looking in the Old Testament you find this: Judges 3:7 - "But he said unto me, Behold, thou shalt conceive, and bear a son; ..... for the child shall be a Nazarite..."
Fulfilled prophecy? No. Just another example of a New Testament writer not having any understanding of the words they were plagiarizing. A "Nazarite" is someone who doesn't cut their hair, not a resident of Nazareth. In fact, archeologists have confirmed that the city of Nazareth didn't even exist at the time that Jesus was said to have lived.
The Gospels are riddled with these types of "errors" regarding alleged prophecies. I could go on and on....
No. The New Testament was tweaked to make it look as though he had.
You misunderstand the Jewish concept of a messiah. For one thing, the Messiah is not supposed to be a divine being, a son of God, the very concept of which is blasphemous in Judaism. Jesus did not conform in any way to the Jewish understanding.
Most Christians didn't get it either, and they didn't grasp the contexts in which various OT books and verses were produced, which is why they tried so hard to make it look as though Jesus was predicted by the Old Testament. To Jews this was just flat-out ridiculous.
a) The Moshiach will be a man of this world, an observant Jew with "fear of God" (Isaiah 11:2)
b) He will be descended from King David (Isaiah 11:1) via King Solomon (1 Chron. 22:8–10)
c) Once he is King, leaders of other nations will look to him for guidance (Isaiah 2:4)
d) He will be a messenger of peace (Isaiah 52:7)
e) Evil and tyranny will not be able to stand before his leadership (Isaiah 11:4)
f) He will include and attract people from all cultures and nations (Isaiah 11:10)
g) The Temple will be rebuilt (Ezekiel 40) resuming many of the suspended mitzvot, “On that day the Prince is to provide a bull as a sin offering for himself and for all the people of the land.” (Ezekiel 45:22)
h) He will then perfect the entire world to serve God together (Zephaniah 3:9)
i) "For My House (the Temple in Jerusalem) shall be called a house of prayer for all nations" (Isaiah 56:3–7)
j) The peoples of the world will turn to the Jews for spiritual guidance (Zechariah 8:23)
k) The whole world will worship the One God of Israel (Isaiah 2:17)
l) Knowledge of God will fill the world (Isaiah 11:9)
m) Nations will recognize the wrongs they did against Israel (Isaiah 52:13–53:5)
n) All Israelites will be returned to their homeland (Isaiah 11:12; Jeremiah 33 :7-9)
o) Jews will know the Torah without having to study (Jeremiah 31:33)
p) The Jewish people will experience eternal joy and gladness (Isaiah 51:11)
q) The ruined cities of Israel will be restored (Ezekiel 16:55)
r) The Sanhedrin will be re-established (Isaiah 1:26)
s) He will take the barren land and make it abundant and fruitful (Isaiah 51:3, Amos 9:13–15, Ezekiel 36:29–30, Isaiah 11:6–9)
t) Death will be swallowed up forever. There will be no more hunger or illness, and death will cease (Isaiah 25:8)
u) All of the dead will rise again (Isaiah 26:19)
v) Weapons of war will be destroyed (Ezekiel 39:9)
w) "He shall not fail or be crushed until he has set the right in the earth." Isaiah 42:4
The Messiah: a-h, w
The Nations: h-m
Israel: g, m-r
The Messianic Era: s-w
Jesus didn't fulfill ANY of those. The ones Christians say he DID fulfill prove nothing other than they can take a verse out of context or change the words around to impress those who don't know what the original said or meant.