There is no historical record before the end of the Second Age that mentions the Morannon specifically, so it isn't possible to state precisely how long it had existed. However, there is a reference in Akallabêth that tells us that Sauron '...had fortified the land of Mordor and had built there the Tower of Barad-dûr...' between about II 1000 and II 1600. It seems reasonable to consider the Morannon as part of this fortification, which would mean that it stood for most of the Second Age, as shown in the timeline above. At the very least, we know it was in place by the end of that Age, because it is mentioned in accounts of the War of the Last Alliance.

It is not known what happened to the Morannon during the Third Age, before its final destruction in the War of the Ring. It may have been destroyed after the War of the Last Alliance, or incorporated by the Gondorians into their own defence of Mordor. An account of the Battle of the Camp in Unfinished Tales mentions the Morannon several times - this took place in III 1944, so the balance of evidence seems to favour the idea that it remained in place during the Third Age. Whether Sauron needed to rebuild it or not, it was certainly in place and under his control at the time of the War of the Ring.