What did Muslims think about the year 1591-1592 (1000 in the Muslim calendar)? Did they worry about an apocalypse? Did they think there would be a new golden age of Islam?

Yes, many Muslims in the Ottoman Empire did worry that there was going to be an apocalypse. The time period from roughly 1585-1610 was one of crisis for the Ottomans, and the numerous uprisings in Istanbul and in Anatolia were placed into this eschatological worldview as a sign of the coming end of the world. Even after it became clear that there would be no apocalypse, the year 1000 continued to be seen as significant, and was made the starting point for several works of history written during the seventeenth century, such as those of Katib Çelebi and Mustafa Naima. The scholars and intellectuals that we know of generally had a negative understanding of history, imagining that the empire was in decline and that the course of history was for the world to (spiritually) decay and worsen as the age of the Prophet Muhammad drifted further into the past. Their predictions for the second Islamic millennium would have been largely negative, not positive.

Cornell Fleischer's Bureaucrat and Intellectual in the Ottoman Empire: The Historian Mustafa Âli, 1541-1600 (Princeton: Princeton University Press, 1986) should have a lot to say on this topic during the relevant period of his subject's biography, though unfortunately I don't have the book on hand at the moment.