Revelation 3.12 The one who conquers, I will make him a pillar in the temple of my God. Never shall he go out of it, and I will write on him the name of my God, and the name of the city of my God, the new Jerusalem, which comes down from my God out of heaven, and my own new name.
Revelation 21.1-2 Then I saw a new heaven and a new earth, for the first heaven and the first earth had passed away, and the sea was no more. And I saw the holy city, new Jerusalem, coming down out of heaven from God, prepared as a bride adorned for her husband.
Revelation 21.10 And he carried me away in the Spirit to a great, high mountain, and showed me the holy city Jerusalem coming down out of heaven from God.
Here, John describes the city New Jerusalem, consistently describing it as coming from heaven. To earth, I would assume. 
Galatians 4.22-26 For it is written that Abraham had two sons, one by a slave woman and one by a free woman. But the son of the slave was born according to the flesh, while the son of the free woman was born through promise. Now this may be interpreted allegorically: these women are two covenants. One is from Mount Sinai, bearing children for slavery; she is Hagar. Now Hagar is Mount Sinai in Arabia; she corresponds to the present Jerusalem, for she is in slavery with her children. But the Jerusalem above is free, and she is our mother.
Here, Paul takes the historical account of Hagar and Sarah, and presents it as an allegory, in which Hagar represents the "present Jerusalem", and Sarah represents "the Jerusalem above".
Based on the context of the Galatians passage, Paul states "this present Jerusalem" as being "in slavery with her children". But he contrasts this with "the Jerusalem above", which "is free". Contextually, I think it's very clear he is contrasting the Old Covenant ("this present Jerusalem", which, indeed, was still "in slavery with her children", clinging to the Old Covenant) against the New Covenant ("the Jerusalem above").
John and Paul received their prophetic knowledge from the same source (God), so it seems reasonable to me that "the New Jerusalem" is the same thing as "the Jerusalem above".
Hebrews 11.16 But as it is, they desire a better country, that is, a heavenly one. Therefore God is not ashamed to be called their God, for he has prepared for them a city.
It is not known for certain who wrote Hebrews, but its writing and verbal mannerisms have been considered to be "Pauline" in style. Here the author speaks of Christ's followers desiring "a heavenly" home, and goes on to describe how God "has prepared for them a city". This is very close in essence to the statements found in the Revelation verses above, in which John repeatedly states that the New Jerusalem was "coming down out of heaven from God". Paul's statement in Galatians, then, is bridged by this verse in Hebrews to John's statements in the Revelation.
I posit, then, that Paul and John are referring to the same city (I'll go ahead and just call it "New Jerusalem", since it distinguishes it from the earthly city), as is the author of Hebrews. But my main point comes next.
Hebrews 12.22 But you have come to Mount Zion and to the city of the living God, the heavenly Jerusalem, and to innumerable angels in festal gathering...
Here, in Hebrews 12.22, the author directly states that his contemporary followers of Christ "have come" already "to the city of the living God, the heavenly Jerusalem". John depicts the New Jerusalem, as coming from out of heaven from God (i.e., it comes from "above"). Would this not mean that John's "New Jerusalem" was a present reality for the first-century author of Hebrews, when he said "you have come to ... the heavenly Jerusalem", if indeed "Jerusalem above", "the heavenly Jerusalem", and "the New Jerusalem" are the same thing?
I cross-posted this topic on another forum, but since it is adamantly anti-[full] Preterist, I had to finish at this point, and only asked others a few questions on what they thought. However, since we are free here, I'll go ahead and put my full thoughts in.
To me, it seems abundantly clear that Scripture teaches that the New Jerusalem (Jerusalem above, the heavenly Jerusalem), which John states is a part of the new heavens and new earth, was already becoming a present reality for the first-century Christians, near the end of the "this generation" that Christ prophesied about. The author of Hebrews directly says that his contemporaries "have come" to "the heavenly Jerusalem". Young's Literal translates it as "you came", but the other 17 versions I have have a consensus of "you have come". Regardless, both cases place the event in the first century. This would necessarily mean that the New Jerusalem is not so absolutely literal as many claim, and Revelation 21-22 was already on its way to fulfillment in the first century, not some future fulfillment.