This question is reminiscent of the Olivet Discourse in Luke 21:5-37 (cf., Mt 24-25). In Luke 21 Jesus has offered a stinging critique of the Temple and its leadership and walked out of the Temple through the east gate to the Mount of Olives. While walking through the beautiful buildings and gate, Jesus predicts they will be destroyed. At least some of the disciples ask at that time about the timing of this event – is Jesus about to restore the kingdom, perhaps judge the current corrupt priesthood and replace it with a pure priesthood. This is the same sort of question someone at Qumran might have asked, since they too thought the priesthood in Jerusalem was corrupt and would be replaced by a more pure priesthood (their sect!)
After the resurrection, it was only natural to think that Jesus would now enter the Temple in the power and glory of the resurrection and begin to reform the religion of Israel and begin the process of evangelizing the nations.
Again, this was a clear expectation of the Messiah’s activity. Beginning with the people of God themselves, Messiah would either convert the enemies of Israel or destroy them (depending on their response or the attitude of the writer describing Messiah’s activities!) Very often these enemies were within the nation itself. Individual groups identified the primary enemy of a pure Jewish faith as corrupt priests, people who did not fully keep the law, etc.
The verb that is translated “restore” in this context (ἀποκαθίστημι) is a key eschatological term. It appears in Mal 4:6 (LXX 3:23) and LXX Daniel 4:26, and it anticipates Acts 3:21 where the related noun appears in an eschatological context. The hope of Israel was that the kingdom would be restored to them as the prophets had predicted (Isa 2:2-4; 49:6, Jer 16:15; 23:8; 31:27-34; Amos 9:11-15). In fact, Luke began his first book with the hope of the coming Messiah in the Song of Zechariah (Luke 1:69-74) as well as the words of Simeon in the Temple (Luke 2:24-32).
Jesus reminds them it is not for them to known when the kingdom will be restored, but they are to be witnesses to the good news of Jesus in Jerusalem, Judea, Samaria, and all the earth. To some extent, the kingdom is about to begin in the Temple in a manner which is not unlike what many expected. The Holy Spirit will fall upon people and they will speak the Word of God in power in the Temple itself.
What is unexpected is that the kingdom would be given to a group of Galileans rather than a faction within Judaism (Pharisees, Essenes, etc.) was not expected at all. From the perspective of Second Temple Judaism as we understand it, these people would be the most unlikely group to be the witnesses of the Messiah to Israel and then the rest of the world!
But this “unlikely group” is another example (in Luke/Acts or the whole Bible) of God choosing to accomplish his goals through the most unlikely and weak things of this world. The restoration of the Kingdom begins with the preaching of two Galilean fishermen in the Temple courts, announcing the death and resurrection of the Messiah. Are there other elements of “restoration” in Acts 2-3 we ought to include here as well?