Paul is indeed referring to the Second Advent in the following passage:
20) and times of rest may come to you from before the face of the Lord; and he may send to you him, who was made ready for you, Jesus the Messiah: THE N.T TRANSLATED FROM THE SYRIAC PESHITO VERSION BY JAMES MURDOCK
Paul's address to this Jewish audience brought to their attention concepts they may have been familiar with but not exactly clear on. For instance, the phrase "times of rest" (KJV and NIV renders times of refreshing) is believed by NT scholars to be one and same as what Simeon, the father of John the Baptist, awaited in Luke chapter 2.
Luke 2:25 And there was a certain man in Jerusalem, whose name was Simeon. This man was upright and just, and was waiting for the consolation of Israel, and the Holy Spirit was upon him.
The mentioned "times of refreshing" or consolation is the heavenly reign of the enthroned Christ. As in Peter's address in Act's 2, Paul similarly summons his listeners to repent for crucifying the very One the prophets of old said would come. Of course their understanding of the Messianic Kingdom was lacking. But Paul's message is clear: Repent, and you too will enjoy the promises foretold by the holy prophets.
We currently live in the refreshing time, and it is but a foretaste of Christ's reign in the age to come in which Christ will no longer reign in the midst of His enemies (Psalms 110:2). When the Restoration of all things (please see Acts 1:6) is completed, Christ will be sent by the Father to destroy His enemies establishing his reign in a new heavens and new earth.
Act 3:23 And it shall be, that every soul, which shall not hearken to
that prophet, shall be utterly destroyed from among the people.
As far as which rendering of Messiah/Christ is appropriate, I've pasted A.T. Robertson's notes from his Word Pictures on the subject.
In the N.T. only here and Act_22:14; Act_26:16. It is not “Jesus Christ” here nor “Christ Jesus,” but “the Messiah, Jesus,” identifying Jesus with the Messiah. See the Second Epiphany of Jesus foretold also in 1Ti 6:15 and the First Epiphany described in 1Pe 1:20.