There are theological arguments for rendering this as a reference to the present day, or to the interval between the resurrection of Jesus & the destruction of Jerusalem by Titus, and so on.
But from a strictly theologically neutral standpoint, Isaiah does not indicate how much time will pass, rather, he points out signs by which the fulfilment of the prophecy may be recognized (so a different but related question, perhaps best suited for SE-Christianity, would be what views people hold regarding the fulfilment of the specific details of this prophecy).
Chronological precision is in fact quite foreign to most of Isaiah's prophecies--he (and other Hebrew writers) are known for frequently making a prophetic statement that has multiple applications or multiple fulfilments--such as a first & second coming of the Messiah (see discussion by Ludlow in Isaiah: Prophet, Seer, and Poet pp. 53-54). The Septuagint translators themselves understood this by taking Isaiah 7:14 (one of the most contested passages in all scripture!) to refer not merely to a young woman, but specifically to a virgin.
So if Isaiah made a prophecy that would be fulfilled once ~7 centuries later, and again ~27+ centuries later, it wouldn't make sense for him to give a date.
The Hebrew word of focus here is "acharith", which can be used to refer to the end of all things, but it has a wider semantic range than this. It can also refer to the end of a person's life (e.g. Job 42:12), or just to posterity, which is not bound by a single point in time in the future, and more. It is not on its own a technical term with a single, explicit meaning.
We should be cautious about assuming that anytime the Hebrew אַחֲרִית or the Greek ἔσχατος appear, that they are referring the same, specific time period. Sometimes the "last days" are used by the speaker/author to refer to the present; sometimes the "last days" are used to refer to the future. Acts 3 is an interesting example and I see that a separate question has been raised to discuss it specifically.
Consider some possible usages of the English word "last" (semantically similar but not identical to אַחֲרִית or ἔσχατος):
- I was there in the last week
- I will be there in the last week
In the latter case (pun intended), the last week of what? The last week of the month, the year, the century, all time? Without a predicate, the "last days" can be--in a non-technical sense--the most recent events prior to whenever someone is speaking. In a technical, theological sense, the "last days" has been most commonly used to refer to the time period preceding the second coming of Christ--a time period to which no precise chronological delimitation is ever applied in the Bible.
The interpretation of Isaiah 2:2 that I personally find most compelling is described here.