The paradox concerns the logic by which we are justified in forming our expectations about the future. Suppose we observe a bunch of emeralds and at the time that we observe them, each is green. This evidence is generally thought to support our expectation that on the first occasion on which we observe an emerald after the year 2100, let's say, we will find it to be green. The pattern of reasoning seems to be "If every F [emerald] we have examined has been found to be G at the time we observed it, then we should expect any given F to be G at any future time at which we might observe it." However, this pattern of reasoning (Goodman shows) cannot in fact support our prediction. For suppose that instead of making G = green, we make G = grue, where an object at a given moment is "grue" at that moment if and only if that object is green at that moment and the moment is (let's say) before the year 2100, or the object is blue at that moment and the moment is during or after the year 2100. So every emerald we have examined we have found to have been grue at the moment at which we examined it (since it was green at that moment, and the moment predated 2100). Therefore, by the principle of reasoning I gave above, we would be justified in predicting that the first emerald to be examined after 2100 will be found then to be grue -- that is to say, blue. But we do not regard the greenness (i.e., grueness) of the emeralds we have checked to support the prediction that some emerald after 2100 will be blue (i.e., grue). Therefore, the above principle cannot be the basis of our prediction regarding emeralds.
I see no connection here with the Schrodinger's cat paradox, which concerns the entanglement in quantum mechanics between macroscopic objects (such as a cat) and microparticles (such as a radioactive atom).
With regard to the final paragraph of your question: If we examine an object after T (2100, in my example) and it is green at that time, then the object at that time was also bleen (i.e., blue at that time, if the time was before 2100, or green at that time, if the time was at or after 2100). But for an object to be bleen at a given time (say, after 2100), it does not always have to be bleen. An object that is always green, both before and after 2100, was grue before the year 2100 and bleen after the year 2100. For an object to be bleen at a given moment, it does not have to change color at the year 2100; if the moment is before 2100, then the object is bleen at that moment as long as it is blue at that moment, no matter what it may do after 2100. If it remains blue, then it was bleen before 2100 and later is grue -- just as for an object to be green at a given moment, it does not have to remain green later.
I should caution that Goodman's problem can be posed in terms of various different definitions of "grue" and "bleen". I have used here the definitions that I think pose the problem most easily, but others (including Goodman himself) define "grue" differently for these purposes.