I posted an answer that got a lot of votes. Now the question has been marked a "duplicate", and may be in danger of closing. Would it be alright if I copy and paste my own happily successful answer to the question that is linked as a duplicate of the one I answered first? Also, are these two questions really duplicates?
Simply copying an answer from one question to another would imply that the questions are also exactly the same, which is rarely the case. The system does in fact raise automatic flags where it detects possible duplication/plagiarism.*
Bear in mind that duplicate questions are not necessarily a bad thing as they are linked and that link is shown at both ends in the "Linked questions" panel.
Answering a question bumps it to the top of the "Active" list which can increase traffic and the number of views your answer gets. That may mean that users also raise flags when they realise there are two substantially similar answers. It will probably mean that your second answer will get a higher level of scrutiny (as a late answer) which means that if it is not exactly right for that question and does not add anything useful to answers already present it will get downvotes.
If the questions really are exactly the same and your answer is a good answer to both which would fit in both places unaltered, it suggests that the questions should in fact be merged, which munges all the answers into one place anyway. If this is the case you should raise a flag on the question so that moderators can take a look and carry out that action if they concur.
*Of course, copying your own answer isn't plagiarism, but the system doesn't check who wrote the answers. Duplicating an answer is generally Not A Good Thing To Do. There is obviously a threshold for how much common material raises the automatic flag, but I'm not sure that it's published or how it's calculated.
You are certainly allowed to copy information from one of your own answers to make another answer post. When questions are marked as duplicates, it can often be helpful to take useful information that came up in the answers to the more recent question and put it in an answer post beneath the older question (if it isn't already on that page).
Of course, you should check to make sure that the wording of the answer makes sense for the older question, and that the answer is a good fit. As with other answers, people may vote it up if they think it is useful, or downvote it if they think it isn't useful.