Christ, we are told will rule with a rod of Iron
We are told he will rule in israel
We are told he will rule the world. All nations will worship him
We are told when he rules. Northern and southern kingdoms of israel will be made one again, be back in their land given their fathers. And they too will worship Jesus having repented of the sins of the fathers
What I'm about to mention cannot be tackled without a year of inquiry--less if you have the luxury of full time study.
The NT quotes/alludes to the OT 2500 times. The best official, published list I'm aware of is in the ABS Greet text (Metzger) when the dictionary and references have been included.
The short list of course, is the first 20 times Acts has quotes of the OT, and this material is surely as close (in time) as we can get to the 40 days where 'Moses, the Psalms and the prophets were explained, that the Christ should suffer and be raised in glory, etc.' A translation with notes will have these marked (NIV, NET, etc).
One of the first things you will see in the 20, hopefully, is that a person must realize what the grammar of Acts 2:30,31 is saying about the reign of Christ. It is saying David saw the resurrection event coming, and that is the official enthronement of Christ. Once this is seen, all the 'kingdom of God' expressions fall into place, and the enthroned King concept of Acts 2--4 will also.
That concept is that his kingdom is
imperative; it is
what should be. It does not mean you can pick up the newspaper then or now and find that all is bliss. In fact, the declaration that Christ's kingdom is imperative for all mankind only makes the shameful elites of the world try harder to resist, says Acts 4, quoting Ps 2. This is why all the declarations of Christ and the apostles have that 'almost here, but not quite' feel about them: it is near, at hand, upon you, among you, etc, etc., etc. And why there is the rebuke in Acts 1 about it.
If the apostles meant to detail a future 'reign in Israel,' they sure messed up. Christ even told them to stop thinking about it in Acts 1. In Acts 26, Paul says that Israel is still (in the early 60s) thinking a national kingdom messiah is coming, instead of that the resurrection was already Christ's enthronement for all men. He actually says we (Christians) declare
nothing beyond than 'Christ was to suffer and then be proclaimed throughout the world'. I see no notes there that 'fix' the 'problem' of a kingdom for Israel. (This is why I have a thread here called the 'three lost verses of Acts'--three places where, if a nation-race kingdom was supposed to come, it should have been explained.)