I give this as a separate answer since I have made too many comments already on the existing answer. This summarizes what I take away from the discussion with Henning.
Original question: "Is there a well-ordered uncountable subset of real numbers."
Clarified question: "Is there a well-ordered uncountable subset of real numbers (using the usual ordering on the reals)."
Interpretation 1 of original language: The question asks about existence of an ordering, so it appears we get to choose both an ordering and a subset. Well, the axiom of choice implies there exists an ordering for which the reals themselves are well-ordered. Then, we can just take a subset of reals being the full set itself. Done. This question is kind of stupid. The only structure of reals here that is used is that the set of reals is uncountable. We could repeat the same question with any uncountable set of objects.
Interpretation 2 of original language: Let's suppose we are forced to use the original ordering of the reals. So we are only allowed to choose a subset. The structure of the problem now is such that the problem is interesting. We must use both the property that the reals are uncountable together with existing properties of the usual ordering on the reals. This is likely the correct interpretation because it is the only one in which the problem is interesting.
Observation: On an exam, there is not time to solve the problem two ways and then try to interpret the problem in a way that is interesting. I would naturally assume "interpretation 1" and then I would be quite confused why the exam is asking such a weird question.
I usually view sets as existing independently of temporal concepts (i.e., “outside of time”). The past-tense language used by Henning in comments above is helpful and is consistent with his interpretation of the problem (which is interpretation 2). Interpretation 2 is likely the one intended by the person who designed the question. However, it is not the only interpretation. In fact, interpretation 2 never even occurred to me until the comment-discussion with Henning. I would have gotten this question wrong on the exam, not because it is a hard question, but because I interpreted the question differently.
Thus, it would have been nicer if the original question emphasized that we should use the usual ordering on the reals, so we are only allowed to choose a subset. One uses the usual ordering on the reals almost always, but one can consider different orderings when set theory questions about “well orderings” arise.