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Wikipedia, in "New riddle of induction", sets out Nelson Goodman's paradox as follows:

Goodman defined grue relative to an arbitrary but fixed time t as follows: An object is grue if and only if it is observed before t and is green, or else is not so observed and is blue. (note omitted)

In Goodman's definition, does the observer know whether Time t has passed? It would seem that the observer could not know. Assume t is known. If t has not passed, the timing leaves open the possibility that the object is grue, but allows the observer to simply wait until the clock reaches t and answers the question; and if t has arrived or passed, the open possibility is closed and the question resolved.

Has any analyst answered this? At the beginning of observations, does the Observer know Time t?

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Goodman's "observer" is just any one of us, in our everyday inductive projections. So you can answer your own question by simply reflecting on yourself: do you know whether time t has passed? Do you wait until time t (whatever it is) has passed, in order to choose between "grass is green" and "grass is grue(*)"?

More to the point, what question would waiting solve? That of choosing between "grass is green" and "grass is grue"? But this is not Goodman's question. Goodman takes it for granted that we would always prefer "grass is green" over "grass is grue", without any waiting, and that we will even regard the grue option as absurd. The question is why. Because as far as truth and knowledge are concerned, the two options green and grue seem perfectly equivalent. This, in summary, is the new riddle of induction.

(*) grue = green until time t, blue afterwards

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The observer knows the time t, or to be more precise, it was stipulated and the observer knows it because the definition of grue is well-known.

Otherwise the paradox wouldn't even work. The whole point of the paradox is that by using induction, the observer arrives at nonsensical propositions like "Grass is grue."; if he didn't knew the meaning of grue, including the exact time t, this would not be possible.

The formulation "arbitrary but fixed time t" is just to emphasize that nothing interesting needs to happen at time t, it can be any time in the future.

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You have noticed something that is almost the right question to ask but not quite. The answers point out that Goodman stipulated that the time t was in the future. But stipulations of this sort generally don't make sense. There is no explanation for why you would only be able to settle the grue question in the future. The first problem is that if something is grue there is an explanation for why it is grue and that explanation will be independently testable and criticisable when you compare it to alternative theories that say the item in question is blue or green or whatever even before the change. You could find out whether the item is grue by doing those tests. But any such explanation will specify that the item will turn grue under some specific kind of circumstances that will not refer explicitly to time since the laws of physics don't change over time. So the whole problem would never arise in any real scientific controversy.

This problem serves to highlight the most important reason why the whole idea of induction is complete bunk that has nothing to do with how knowledge is actually created and tested. What tells you whether something will turn green or whatever is an explanation: an account of how the world works. No such account can follow from any set of experimental data, or be derived from experimental data in any way. Experimental data can only be understood in the light of explanations, so there is no such thing as starting with raw data. Rather, you invent new ideas to solve problems with your ideas about how the world works and test them with criticism and experimental data. Induction is unnecessary and doesn't solve any philosophical problems or any other problem. See "Realism and the Aim of Science" Chapter I and "Objective Knowledge" Chapter 1 both by Popper. See also "The Fabric of Reality" Chapters 3 and 7 and several chapters of "The Beginning of Infinity" both by David Deutsch.

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