Not only did they have prophets telling them the time of the coming of the Messiah, ie (Daniel 9:24), but you had a levite priest, John the Baptist's father Zacharias, publicly prophesying that his son was going to declare who the Messiah was. (Luke 1) Then you had the strange circumstances associated with John's birth that would and should have alerted the Jewish inhabitants that something unusual was happening.
Herod heard the rumors, and bringing learned men before him, knowing the signs, narrowed it down to boys under 2 years old in Bethlehem. Shepherds in the field heard of it. The Magi, from a distant land, heard of it.
Simeon and Anna, a prophetess, (Luke 2) prophesised publicly over him in the temple, one or both having asked the Lord to show them that day.
The Jews kept meticulous records of births, etc. and calculated events by generations. As is indicated, also in Luke, in Daniel and other places, a certain number of years willl occur before the time of some important event. If the magi (magicians) could figure it out...
At 12 years old, Christ was in the temple teaching the rabbis, so that everyone wondered about him. In addition, later on, Jesus told them who he was. (Luke 4:18) He confronted the pharisees with a question that they couldn't answer: (Luke 20:41-44.) In addition, the Lord has told us many things about his 2nd coming, so we will have no excuse either when it happens.
As for Judas, the bible says that Satan entered inot him, so Judas must have given him an opening.
The disciples believed on him, as did many many others, but they were people, like ourselves, with lots of responsibilities, and let's not forget, both a fear of their leaders and of the occupying army of the Romans.
Moses said in Deut 4:34 that no one but God Himself can accomplish such a feat, and with signs and wonders. That alone, in retrospect, knowing that Jesus took a nation from God's nation (all that the Father had given him) should be enough now to cause the Jews to believe.
Their idea of a messiah was very different from what it turned out to be (which is why they rejected it and reject it to this day). Messianic Judaism was of a relatively recent origin, preceding our Lord's appearance on earth. It was mainly political and not religious.
BTW the Scripture is confusing about looking for sings in the heaven and certainly the magi were not considered anything but sorcerers in Judaism.
The disciples believed on him, as did many many others, but they were people, like ourselves, with lots of responsibilities, and let's not forget, both a fear of their leaders and of the occupying army of the Romans
What responsibilities did the Apostles have? And if they believed they were in presence of God, what fear would they have shown. Once they became believers (at Pentecost) they lived their faith and died martyrs' deaths for it. before the crucifixion, they scattered like scared little rabbits!
I do not see any evidence that they believed except as lip service. as I said, even St Peter sank when given the opportunity to show his faith.