For Jews living in the day, they may well have known. They were very much expecting the messiah (in large because of Daniel's 70 sevens was about due and also because of the Roman oppression), and they would've know their OT well. So quite possibly, they should have known.
But a big part of the problem though, I think, is they were looking for the wrong messiah. The OT describes conflicting views on the messiah, so much that the Jews had determined there must be two. An all-conquering messiah, they called 'messiah ben David'; and the weak and sorrowful messiah, they called 'messiah ben Joseph', and they presumed the all-conquering one was the messiah to come in and save them from their enemies; and the lowly messiah they presumed to be talking about Israel themselves, because they had suffered so much.
But to answer your question, I think you couldn't do it without the NT, because that is where we have the explanation of how he fit the bill. The OT has all the prophecies of what the messiah would do, or suffer, but without the NT to make those points plain, a lot of them would go unnoticed. The gospels are a record of what Jesus did, but crucially it also ties that to the OT prophecies. Without that, many of the prophecies would not be obvious they were even prophecies. A lot of prophecies would have had a fulfillment in their day, but we now see a second, greater, fulfilment.
For example, (and I'm just going by memory, so the wording will be out)..
- God leading Israel out of Egypt as 'my only son, Israel' but now we know this is reference to Jesus returning from exile as a child. Unexpected with the OT alone.
- David's psalm 'you did not let your holy one see decay' - he was writing about himself! David was the anointed. David was the king, and God's holy one. And it applied to David at the time (he was terrified of Saul hunting him and thought he was about to die, but thankful God saved him from that) - so it had a fulfilment at the time, but a greater fulfilment in the Lord.
- 'a sign to you, the virgin (maiden) shall bear a son' - this would have been literally fulfilled in that time, but it would've been a maiden who then had relations with a man and had a child, which was proof for the people that God's word was going to come true (something about the result of impending war conflict, i think). But who would've known it had a greater fulfilment in Jesus conception and birth.
And there are many more.
And all this is why Jesus said to 'forgive them for they know not what they do' when he was crucified.
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