Undergate

Given that the tunnel in which the Undergate stood led to the Tower of Cirith Ungol, it would seem reasonable to imagine that the gate was made after that Tower was raised (at the end of the Second Age), and after it had been abandoned by its original builders, the Gondorians. Gondor's watch on Mordor was withdrawn after the Great Plague of III 1636, but the Orcs would not take over the Tower for some time after that. The Nazgûl captured Minas Ithil, which guarded the way to Cirith Ungol, in III 2002, and after that time the Orcs could more easily have occupied the Tower of Cirith Ungol and constructed their hidden gate.

This reasoning suggests that the Undergate was made in the latter part of the Third Age, but this is not necessarily the case. In his original drafts for The Two Towers, Tolkien wrote that the tunnels of Cirith Ungol had been made by the Orcs in the Dark Years (that is, during Sauron's original rule as Lord of Mordor in the Second Age). This is clearly not impossible, but it is hard to understand why the Orcs would have made a tunnel leading to a tower that had not yet been built. This is perhaps the reason that Tolkien abandoned his original wording, preferring instead to merely describe the tunnels as having been made in an unknown 'far-off time' (The Two Towers IV 10, The Choices of Master Samwise).

Strictly speaking, there were two gate-like structures within the Under-way that led from Shelob's Lair to the lower levels of the Tower of Cirith Ungol. The first of these was a simple block of stone that held the passage within the Lair that led towards the Tower, preventing Shelob herself from using that tunnel. Beyond this block, the passage led on to the Tower itself, where a solid pair of brass doors guarded the way into the deep chambers beyond.

Our references on this matter are ambiguous, and none directly state whether the 'Undergate' was the block of stone or the doors opening into the Tower (or perhaps even both in combination). However, only one of these - the brass doorway at the end of the tunnel - is expressly described as being a 'gate', and on that basis we assume here that the Undergate was the gate into the Tower, and not the stone block that guarded the entrance to the Under-way.

(The relevant comments comes at the end of The Two Towers IV 10, The Choices of Master Samwise. After the Orcs pass in through the metal doors beneath the Tower, those doors are shut and bolted behind them, leaving Sam Gamgee alone in the passage. At that point we're told that 'The gate was shut', implying that this doorway was indeed the Undergate of the Tower of Cirith Ungol.)