On Prophecies Related to the Time-Table of the Messiah's Coming

I'm not sure we have enough information to determine the date that the Messiah would be born however I think we can get to the week of the crucifixion.

Let us assume that the “sevens” are seven. Gabriel told Daniel that after the decree to rebuild, there would be “seven sevens” (which is 49), plus “sixty-two sevens” (which is 434). After these 483 years, the Anointed One would be cut off. If the prophecy is true, this would be the year of the crucifixion. Gabriel said to count “from the issuing of the decree to restore and rebuild Jerusalem.” When was that? The prophet Nehemiah records such a decree, and he dates it as the twentieth year of Artaxerxes. On our calendar, that date is 444 BC. Counting 476 years from 444 BC, and remembering that there is no year numbered “zero” AD, we discover what Gabriel told Daniel: the Messiah would be cut off in 33 AD.

Numbers 24 contains a prophecy that could confirm the messiah's arrival at the very time that magi found his Star. (See below) But first:

The Star of Bethlehem

The quote in the OP should be read to understand that "the time had come," refers to the coming of the Star, not to prophecy. In the account given by Matthew, there is no prophecy about the timing. The timing is about the Star, not prophecy.

When Herod the king heard this [about the Star], he was troubled, and all Jerusalem with him. And when he had gathered all the chief priests and scribes of the people together, he inquired of them where the Christ was to be born. So they said to him, “In Bethlehem of Judea, for thus it is written by the prophet:

‘But you, Bethlehem, in the land of Judah, are not the least among the rulers of Judah; For out of you shall come a ruler who will shepherd My people Israel.’ ” (Mt. 2)

The probable reason that no prophecy about timing is mentioned is that no such prophecy was known. But see below for one suggestion.

Numbers 24

The OP is looking for prophecies that might be understood as suggesting the time was ripe. Here is one, and it dove-tails nicely with the Star of Bethlehem. It is the prophecy of Balaam in Numbers 24:17-19

I see him, but not now; I behold him, but not near. A star will come out of Jacob; a scepter will rise out of Israel. He will crush the foreheads of Moab, the skulls of all the people of Sheth.

It seems strange that Herod's advisers missed this, since the Star figures so prominently in the narrative. It was not missed a few generations later, when the renowned Rabbi Akiva used it to describe the messianic leader Bar Kochba, who led a temporarily successful revolt against Rome in the mid 2nd century.

I suggest that a reason this prophecy is relatively unknown among Christians is that it tends to support the idea of the Messiah as a war leader, as Bar Kochba was, rather than as a spiritual redeemer.

Conclusion

The clearest prophecy that might have predicted/confirmed the timing of the Messiah's birth is probably Numbers 24:17-19. However, this scripture does not seem to have created any particular stir among the people of Jesus' time.

An intriguing aspect of all biblical prophecies about future events is the lack of clarity about precise dates when something will happen. Yet enough detail is given so that - with hindsight - people can see that what was foretold did happen at just the right time. This annoys an awful lot of people. They think the Bible ought to give sufficient clues so that individuals can work out in advance when something is going to happen.

Yet Jesus has told us that "It is not for you to know the times or the seasons, which the Father hath put in his own power." Acts 1:7. That was in response to the disciples asking the resurrected Christ if he was going to restore the kingdom to Israel at that time. Their question was wrong because they had a wrong expectation about how the kingdom of God was to display itself, and that led to the hope that the timing of that event was imminent. They were mindful of Jesus' predictions about future events (as per. Matthew chapter 24 and Luke chapter 21). But because of becoming fixated on WHEN foretold events would occur, they had failed to grasp WHAT was to occur in the build-up. Had they pondered Jesus' prophecy, they would have known that many signs he gave were of things that had to happen before his return.

That was the same problem that prevented huge numbers of God's people becoming aware of the time for the foretold Messiah to appear. Prophecies in the Old Testament gave all the necessary clues, such as a forerunner to prepare the way of Messiah. That forerunner was John the Baptist. Had they paid attention to him, they would have been prepared to receive the Messiah when he followed shortly after. Then, when Messiah appeared, they would have known it.

Same with future prophecies we tussle with, such as those in the book of the Revelation. End-time events are described in symbolic language and people get knotted up in when this, and when that should happen. All with a view to second-guessing the date of Christ's sudden appearing. We are not meant to know! We are meant to be prepared for that event at any time, so that even if it does not occur in our lifetime, we will have stayed strong in faith, resisting all the wiles of Satan which are designed to deceive Christians and to dishearten them. We have been categorically told that no man knows the day or the hour for that future event Mat.24:36. Same with Christ's first coming. It wasn't working out in advance a year or a month or a day when Messiah would be born on earth that mattered. It was recognizing HIM that mattered. And he had to be here first before we could identify him. And his forerunner had to have come in advance to prepare a people for him. Those were the ones who became baptized, and who then followed Jesus.

With hindsight we will understand how the prophecies were all fulfilled to the nth degree. We can do that now that Christ has come, and gone, and it will be the same regarding his return. We have been given the signs to look out for but we are not told when, exactly, things will happen. Tragically, those who become obsessed with when things should happen seem to fail to be alert to who is to appear. They think that if only they can work out when he will return, they will be sorted. No, they won't. They will be unprepared, for "at an hour you think not, then the Son of man will come" Mat. 24:44.

It was just the same at his first coming. Yet had they known the prophecies, then when John the Baptist appeared first, they would have been prepared by him and recognized the Messiah as having come. The prophetic signs point to the things signified, not dates. Christ is the one signified, and all the 'clues' in the Bible point to him; the 'when' will happen when the Father determines. Only with hindsight will we understand that the 'when' did, indeed, work out exactly as foretold.

The answer to your question is that when the forerunner appeared, then the Messiah would follow shortly after, as per Malachi 3:1, cf. Luke 1:13-17.

Suggested link for more information: https://www.gotquestions.org/Old-Testament-Christ.html

EDIT in response to comment query: Turn to Isaiah 40:1-5 which relates directly to Malachi 3:1 and Mark 1:3-9

"The voice of him that crieth in the wilderness. Prepare ye the way of the Lord, make straight in the desert a highway for our God... And the glory of the Lord shall be revealed..." (Isaiah 10 A.V.)

"The beginning of the gospel of Jesus Christ, the Son of God: As it is written in the prophets, Behold I send my messenger before thy face, which shall prepare thy way before thee. The voice of one crying in the wilderness Prepare ye the way of the Lord, make his paths straight. John did baptise in the wilderness, and preach the baptism of repentance for the remission of sins." (Mark 1)

John the Baptist was that messenger (i.e. angel) in the wilderness, preparing Israelites to recognise Jesus as the foretold Messiah. Those who believed John, repented and were baptised, and they received Jesus in faith. The scriptures interpret those prophesies for us, telling us exactly what they mean. We are to believe what they tell us.