Acts 3:20

And he shall send Jesus Christ, which before was preached unto you:

[And he shall send ...] <Acts 1:1> l. Under this economy of things, he shall send Jesus Christ, that is, the Messiah, to teach people; to redeem them; to save them; to judge the world; to gather his people to himself; and to condemn the wicked. Under this economy they were then. This, therefore, was an argument why they should repent and turn to God, that they might escape in the day of judgment.

[Which before was preached ...] Who has been proclaimed as the Messiah. The name "Jesus Christ" is equivalent here to "the Messiah." The Messiah had been proclaimed to the Jews as about to come. In his time was to be the period of refreshing. He had come; and they were under the economy in which the blessings of the Messiah were to be enjoyed. This does not refer to his personal ministry, or to the preaching of the apostles, but to the fact that the Messiah had been a long time announced to them by the prophets as about to come. All the prophets had preached him as the hope of the nation. It may be remarked, however, that there is here a difference in the manuscripts. A large majority of them read [prokecheirismenon] (grk 4400), who was designated or appointed, instead of who was preached. This reading is approved by Griesbach, Knapp, Bengel, etc. It was followed in the ancient Syriac, the Arabic, etc., and is undoubtedly the true reading.

(from Barnes' Notes)

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For the response to your questions we will be taking into account the context of these verses. We are mainly considering Acts 3:11- 4:2.

Peter sees immediately the opportunity given of God (the people coming to see the lame man which was miraculously healed) to declare unto the Jews "men of Israel" through Jesus the resurrection from among the dead. He makes it clear that it is an issue of Christ being raised from the dead, "The God of Abraham, and of Isaac, and of Jacob" – (The God of the living (Mtt 22:32)), "hath glorified his Son Jesus… God hath raised from the dead; whereof we are witnesses" (v13, 15). And Christ having been raised, "the faith which is by him hath given him this perfect soundness in the presence of you all" (v16). Peter goes on to acknowledge that because of ignorance they knew not the coming of Christ "Messiah" and therefore "ye denied the Holy One and the Just, and desired a murderer to be granted unto you; And killed the Prince of life" (v14-15). But Peter says that this was in fulfillment of what was spoken by the mouth of all the prophets (v18). This being the case, Peter having presented unto them the evidence concerning Christ’s coming unto them (Jn 1:11) and His Death and Resurrection (all in fulfillment of scripture (1Cor 15:3-4)), he commands them to repent – to turn unto the Living Risen Christ so that "the times of refreshing" may come "from the presence of the Lord" (the reality of being in union with the Living Christ, and having no life but Christ).

References:

Mtt 22:32

I am the God of Abraham, and the God of Isaac, and the God of Jacob? God is not the God of the dead, but of the living.

Jn 1:11

He came unto his own, and his own received him not.

1Cor 15:3-4

For I delivered unto you first of all that which I also received, how that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures:

 

Next, starting with verse 20, Peter picks up once again declaring how that the Lord had indeed sent unto them "Jesus Christ which before was preached unto you" (v20). And that the heavens did indeed receive Him (Eph 1:19-22, Heb 10:12-13). Peter goes on to declare by Moses and all the prophets, how that they "foretold these days" (vs 22, 24). "These days" - the time of Christ’s coming in the flesh and consequent Death, Burial, and Resurrection. Peter goes on to bring in another evidence of "these days" with the picture of Christ from the Old Testament as he shows Christ as the fulfillment of "the Seed" in reference to Abraham (Acts 3:25, Gal 3:16, Joh 12:23-24).

References:

Eph 1:19-22

And what [is] the exceeding greatness of his power to us–ward who believe, according to the working of his mighty power, Which he wrought in Christ, when he raised him from the dead, and set [him] at his own right hand in the heavenly [places], Far above all principality, and power, and might, and dominion, and every name that is named, not only in this world, but also in that which is to come: And hath put all [things] under his feet, and gave him [to be] the head over all [things] to the church,

Heb 10:12-13

But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God; From henceforth expecting till his enemies be made his footstool.

Acts 3:25

Ye are the children of the prophets, and of the covenant which God made with our fathers, saying unto Abraham, And in thy seed shall all the kindreds of the earth be blessed.

Gal 3:16

Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ.

Jn 12:23-24

And Jesus answered them, saying, The hour is come, that the Son of man should be glorified. Verily, verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit.

 

Peter next goes on to declare through Jesus the resurrection, and that this resurrection is first unto the Jew "Unto you first God, having raised up his Son Jesus, sent him to bless you…" (Acts 3:26, Rom 1:16). At this point as they spoke the religious leaders were more than upset because "they taught the people, and preached through Jesus the resurrection from the dead."

Reference:

Rom 1:16

For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek.

 

The whole issue in this passage is Christ being the resurrection that Israel waited for. The apostles show that in the Old Testament scripture He is declared to be just that. The mention of "And he shall send Jesus Christ" (Acts 3:20) is in reference to the coming of Christ in the flesh to the Jews, their rejection of Him, and His consequent Death, Burial, and Resurrection. This was indeed a future tense for the Old Covenant saints, "But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law" (Gal 4:4). The apostles show how that God did indeed send His son Jesus Christ "which before was preached unto you" by the prophets etc… . Now, with such evidence of Christ, the apostles call those Jews present to repent and receive Him as their resurrection and Life that they might enter into union with the Living God.

(from C.M.I. Bible Research Center)