Jesus is the Messiah. He fulfilled the old covenant.

The Old Testament comprises the Jewish law (the "Law") and the writings of their prophets. The Law is written in the first five books of the Old Testament (Genesis, Exodus, Leviticus, Numbers, and Deuteronomy). It is the Jewish Torah and mandates obedience to God's Word. The law and the writings of the prophets all foretold of the coming of Jesus the Christ, the Messiah. Jesus summarized all of the Law and the prophets by His Word which He is.

• Jesus said unto him, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind. This is the first and great commandment. And the second is like unto it, Thou shalt love thy neighbour as thyself. On these two commandments hang all the law and the prophets. Matthew 22:37-40.

Therefore, Jesus is the fulfillment of the Old Testament Law and the prophets.

• Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. Matthew 5:17.
• And he said unto them, These are the words which I spake unto you, while I was yet with you, that all things must be fulfilled, which were written in the law of Moses, and in the prophets, and in the psalms, concerning me. Luke 24:44.

By way of reproof, the moment before Jesus died on the cross, He spoke of His purpose to fulfill the Old Testament Law and the prophets as being finished. 

• When Jesus therefore had received the vinegar, he said, It is finished: and he bowed his head, and gave up the ghost. John 19:30.

When we believe Jesus is the savior, we need not study the Old Testament because the Old Testament is the witness to this truth. When we believe, we don't need to study a witness; all we need to do is know it's there. Sure, we may quote Old Testament scripture as a timeless witness to the New Testament scripture, but many Christian churches spend way too much time in the Old Testament when the working of the Spirit by the faith of Jesus in us is from the New Testament, the New Covenant. Some even say the Old Testament can hold people back from growing in the Spirit if they ignore the New Testament because back in Old Testament times, the Holy Ghost did not reside inside the believer like it did Jesus, and now in us, the born again in Christ Jesus.

Yes. The Spirit is available instantly for us to partake of (drink) through the Word of God and shall flow from us if we believe in Him, Jesus the Word. 

• Jesus answered and said unto her, If thou knewest the gift of God, and who it is that saith to thee, Give me to drink; thou wouldest have asked of him, and he would have given thee living water. John 4:10.
• In the last day, that great day of the feast, Jesus stood and cried, saying, If any man thirst, let him come unto me, and drink. He that believeth on me, as the scripture hath said, out of his belly shall flow rivers of living water. (But this spake he of the Spirit, which they that believe on him should receive: for the Holy Ghost was not yet given; because that Jesus was not yet glorified.) John 7:37-39.

Remember, 

• God is a Spirit... John 4:24.
• For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one. 1 John 5:7. 

ALL three parts, collectively and individually are God as a Spirit, the Spirit of God (the "Spirit"). The Father is the Spirit, the Word is the Spirit, and the Holy Ghost is the Spirit. We don't say, Father Spirit, Word Spirit, or Holy Ghost Spirit, but we could. Each part of the Spirit is holy ("Holy"). We could say the Holy Father (definitely not the pope), the Holy Word, and the Holy Holy Ghost. (The Holy Holy Ghost is also called the "Spirit of Truth" in the King James Bible to avoid confusion.)

• And now I am no more in the world, but these are in the world, and I come to thee. Holy Father, keep through thine own name those whom thou hast given me, that they may be one, as we are. John 17:11.
• John 14:17, 14:26, 15:26, and 16:13.

Yes. The King James Bible (the "Bible") is timeless, but the versions aren't. The Bible consistently covers the past, present, and future as it is written from the beginning, but the versions change the Word and install inconsistencies from the time they are written. 

As we study the New Testament, the proper context of the Spirit is revealed and best understood if we appreciate the times in relation to the revelation of God, the Father, the Word, and the Holy Ghost, who are all timeless. 

Please pay special attention to: "Jesus died on the cross" and "Today, after the resurrection of Jesus".

A.  The Word and the Father:

• The beginning and the creation of mankind. The Word of God the Father was given directly to His children, the first man and woman. Knowing He created them having a will of man to decide between the will of the flesh (what they want) and the will of God (what He wants), God the Father told the first couple not to touch (or eat of) the forbidden fruit tree.
• Abraham and the chosen Jewish nation. Beginning with the Ten Commandments the Father gave to Abraham, the patriarch of the Jewish nation, the Word was the law for the Jews. 
• Old Testament times, before Jesus. The Holy Ghost gave utterance (a spoken Word) to the Jewish prophets, but He (the Holy Ghost) did not indwell (reside inside) them. The Old Testament writings of the prophets are also the Word of God. The Old Testament prophets foretell the coming of Jesus the Christ, the Messiah. Back then, salvation was for the Jews only, and there was no hope for the rest of us ("Gentiles").

B.  The Word and the Son:

• Jesus was born the only begotten son of God. The Word of God (Spirit) was placed in Jesus as His spirit of man (the will in Him), making Him a son of man (in the flesh) and a son of God (in the Spirit). As a young man, before His ministry, Jesus was a Jew bound by the law, the prophets, and the Jewish religious practices. Salvation was still for the Jews only.
• The New Testament is the current covenant; the other one is the Old one (not current). 
• Jesus begins His ministry with the Holy Ghost. As an adult, Jesus was baptized for repentance of sin He did not commit and showed us the Holy Ghost shall dwell within the cleansed temple (body) of the penitent. Jesus began his ministry at the leading of the Father as communicated through the Holy Ghost, who bears witness to the Word in Jesus from conception and by the power of the Truth. (The Holy Ghost is also called the Spirit of Truth and referred to as another (second) comforter. All who believed in Him (i.e., His Word) are saved by the Spirit of His Word and the power of the Spirit of Truth witnessing it.
• Jesus died on the cross. The death of Jesus on the cross washed us of our sins if and only if we believe He is the savior. 

WARNING: When our body dies, our eternal soul will follow the destination of our spirit as it returns to its source. Those who have chosen to remain unregenerate and not renewed (born again) through repentance by the Spirit of God will die forever in the torments of hell, the opposite of paradise as their spirit joins their father, the enemy of God, the devil. 

C.  The Father, the Word, and the Holy Ghost:

• Today, after the resurrection of Jesus. After Jesus was raised from the dead, He delivered the Holy Ghost from the Father to indwell all of His disciples individually, then and now. Yes, including us today. His disciples, then and now, are those who conduct their lives seeking to do the will of the Father by the guidance of the Holy Ghost we received through repentance. Then and now, Jesus commissions His disciples, the real Christians, to do the works He did as His ambassadors. 
• Now then we are ambassadors for Christ, as though God did beseech you by us: we pray you in Christ's stead, be ye reconciled to God. 2 Corinthians 5:20.
• And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth. Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you always, even unto the end of the world. Amen. Matthew 28:18-20. 

(Note: If anyone ever tells you the Holy Ghost is not for us today, they do the work of the devil. That's also why the "versions" of the Bible hide the Holy Ghost contrary to the original scriptures.)