QUESTION:
  How do we know that Yeshua was the Messiah that was spoken of in the Old Testament? 

ANSWER:
  The greatest proof that Yeshua was the Messiah comes from Old Testament prophecies themselves.  If something VERY SPECIFIC is mentioned hundreds or even thousands of years before the event actually happens, that would be astounding proof.  However, if hundreds of such prophecies were fulfilled without a single error, we must then conclude that our proof is both irrefutable and supernatural. 

Thus, let us now review just one of the hundreds of prophecies Jesus fulfilled by His first coming approximately 2,000 years ago.

In the book of Daniel, written hundreds of years before Yeshua, it says: "Know therefore and understand, that from the going forth of the commandment to restore and to build Jerusalem until the Messiah the Prince shall be seven weeks, and threescore and two weeks: the street shall be built again, and the wall, even in troublous times" (Daniel 9:25).

Thus, the Bible teaches the ruler (the Messiah or Yeshua) will come 69 (62+7) weeks of years[1] after the decree to rebuild Jerusalem.  Jerusalem was to be rebuilt with streets and a trench (or wall) and was a major reconstruction.  In addition, we know that a  prophecy year was 360 days which matched the Babylonian calendar year.  When Daniel wrote the book named after him, Israel was captive to the Babylonians, so this does actually make sense. 

How was the King to be recognized?  Zechariah says: "Rejoice greatly, O daughter of Zion; shout, O daughter of Jerusalem: behold thy King cometh to thee: he is just and having salvation; lowly, and riding on an ass, and upon a colt, the foal of an ass" (Zechariah 9:9).

So, who rode into Jerusalem on an ass and what did all the people proclaim?  "And they brought him to Jesus: and they cast their garments upon the colt, and they set Jesus thereon.  And as he went, they spread their clothes in the way.  And when he was come nigh, even now at the descent of the mount of Olives, the whole multitude of the disciples began to rejoice and praise God with a loud voice for all the mighty works that they had seen; Saying, Blessed be the King that cometh in the name of the Lord: peace in heaven, and glory in the highest. And some of the Pharisees from among the multitude said unto him, Master, rebuke thy disciples. And he answered and said unto them, I tell you that, if these should hold their peace, the stones would immediately cry out"
(Luke19:35-40)

This is the only time Jesus allowed Himself to be presented as "King".  Before this event, His time had not yet come.  See Yeshua's Millennial Kingdom.  God's faithfulness to His Word was never lacking, "...I tell you that, if these should hold their peace, the stones would immediately cry out." 

See the following commentary page: Daniel 9:24-27 

In addition see: Jewish Bible Contentions - Isaiah 53

Also see Messianic Prophecies List