When did it get built?
How did it get built?
Why isn't there any mention of it in The Revelation of Jesus Christ?
And the reporter would be right. Is that any way for the Bible (and/or God) to be ... not to lay it all out and answer the "who, what, when, where, and why" -- right there in the article (at least the chapter or the "book" you're looking at)? With the Bible, things are "scattered" in different books and not all things are told to us, unfortunately.
BUT..., what we find out, with God and the Bible, is that even though all things which are told to us in the Bible, including associated facts and other "information" is 100% inerrant and infallible -- we do not have a "history book" (in the normal sense of a history book).
If we did have a history book (even though "history" is given to us in the Bible), we would have those answers laid out for us in very good and orderly fashion.
The Bible is not a history book in that it's going to give us those things that we normally expect to see in history books. It does give us "history" -- but only as it's relevant to what God is saying to us or as we "get it" in a tangential sense.
Anyway, I just had to say that, first... :-)
So..., to your questions... You were asking ...
When did it get built?
How did it get built?
Why isn't there any mention of it in The Revelation of Jesus Christ?
I take it that these questions are related to the Third Temple and not to Ezekiel's Temple.
All you can say about when it gets built is before the midpoint of the Tribulation, and far enough back from the midpoint of the Tribulation so that there is time for the construction to be completed.
I've read that the building of it could be completed in 12-18 months and be ready for operation, given that plans are ready to go (or just about ready to go) and materials have been gathered (or are in the process of being gathered now). So, it's like a whole construction crew, sitting on the sidelines, all materials ready and waiting, and they're just waiting for the signal to "Go!" -- and then they build it as fast as they can... :-)
Thus, the Temple gets built anywhere from now to about 12-18 months before the midpoint of the Tribulation. The reason why the "timing" for the building of the Temple is not given in the Bible -- is -- because of the Rapture being a "signless event".
If one were told that the Temple was going to be built a year before the Tribulation begins... you would have a "sign" for the timing of the Rapture. Since no time is given for the building of the Third Temple... that keeps the Rapture from being "dated" (as so many people try to do... :-) ...).
And continuing... as to how it gets built. I presume you mean how is it possible that it can be built, given the way the Muslims are about it right now. I don't believe you're talking about "construction details" and "floor plans"... :-)
Again, I think the lack of detail here, is related to the "imminency of the Rapture".
I think if God gave more detail here (and also, perhaps this is detail that is simply not necessary, too) ... then it would give some kind of "dating" for the Rapture.
You can see that I'm "keying" a lot of this into the idea that when the Rapture happens is not going to be known ahead of time, and thus, if we get more detail about some of these things, it would make the Rapture an event that could be dated, beforehand.
Why the Muslims would permit such a thing to happen... well... the answer to that one may be related to Psalm 83 and Israel's victory over the surrounding Muslim nations and having Israel "living in peace" at the time (because that's what the Gog/Magog war says, in Ezekiel 38/39).
And lastly, why isn't it mentioned in Revelation? Well, I've got to detour here a little bit. First of all, the Bible is an integrated whole and God gave us His "progressive revelation" (the Word of God) over a long period of time -- which to God was "nothing" (in the manner of figuring time). The entirety of mankind is a mere "blip" on God's timescale, and thus, God gave us His full revelation, over a period of a few millennia, which to Him was like sitting down at a desk and writing it out for us... just a "moment of time" in God's framework. To us, it's a long time, but to Him, it's nothing, and He did mean for it to be "progressive", too... revealing more and more until the fullness of Revelation was given, and the canon of Scripture closed.
Okay, so in that light, having something in Revelation or not, means nothing, actually. It's all one book, given to us by God, and Revelation is simply the end of the book (of the Word of God).
BUT..., having said that, the Third Temple is mentioned in Revelation... chapter 11.
Now..., aside from the Third Temple, the Millennial Temple that is spoken about in Ezekiel, is going to require a lot of things be changed, including the topography of the land, plus the boundaries of Israel and that appears to happen at the end of the Tribulation and at the beginning of the establishment of the Kingdom of the Messiah, here on earth.
You probably already knew all that... but it's always good to go over it for anyone who is reading here.
If I didn't quite get it right, for what you were asking... let me know ... or discuss some aspect of what I was talking about. :-) I hope this helps.