So . . .
Well, if you said “Yes,” you are on the right track, but why? Your reason for saying so matters a lot. At the end of this post the “real answer” is provided. If you said “Yes, it is correct” simply because the phrase is so familiar that it “sounds okay,” you are on dangerous footing. Consider four points:
Point Number 1: The “sounds okay” approach to grammar is perilous, especially in these days. Incorrect phrases can be heard so often that they have acquired a kind of presumptive legitimacy. But that still makes them wrong, and it makes the utterer appear ignorant of, or (even worse!) indifferent to, accuracy in English usage.
Point Number 2: Let’s suppose that you said that “Woe is me” is wrong. Why would you say that? Well, Probably you know that “is” (the third person singular form of the verb “to be”) works much like an equals sign. This means that it cannot take a direct object because it does not act on anything. By definition, the subject (“woe”) must be in the nominative case; thus the noun or pronoun on the right side of the “is” must also appear in the nominative case, and not in the objective case. To illustrate, “It is me” is wrong because “me” is in the objective case, and “is” doesn’t have direct objects. It should be “It is I” because “I” is the nominative case of the first person singular personal pronoun. This is an example of the sort of phrase about which the “sounds right” school of thought could quickly seduce you into error.
Point Number 3: But if you believe that “Woe” is properly linked with “I” (not “me”), should it be “Woe am I” or “Woe is I”? Here there are two schools of thought. The first says that the subject is in the third person because all nouns (common and proper) are in the third person. Therefore the verb should be “is” and not “am.” If someone calls on the phone and asks to speak with you by your name (i.e., “Is Mr. Smith there?”), you might say, “This is he.” Or you might say, “It is I.” Both are using “is,” not “am.” (Most likely, you would say something like “It’s me,” which is indisputably incorrect.) Under this school of thought, the choice would be for “is.”
A second school of thought would point out that the sentence really says “I am woe,” and the word order was simply inverted, which is permissible in English. Compare “Happy am I when evening comes.” Since the subject of the sentence is “I” (looking at matters functionally), then the verb should be “am.”
Point Number 4: Patricia O’Connor wrote a book entitled Woe is I. She said she chose the title to point out that only “a pompous twit” would insist on correct usage, since the phrase “Woe is me” has been acceptable in English for generations. She is partially right: The expression has been acceptable in English for at least 800 years. She is also wrong in one respect: “Woe is me” is not wrong; it is correct! “Me” in this case is not the subject; it is also not the direct object; it is in the “dative case,” that usage of a noun or pronoun that names the person to whom or for whose sake the action (or being) of the verb takes place. With action verbs, this is usually the indirect object of the sentence. To understand this, consider the old Vaudeville exchange: “Quick! Call me a cab!” “OK, you’re a cab!” This joke hinges on confusing the direct object “me” with the indirect object “me.” The expression “Woe is me” is a construction, considered archaic nowadays, for “Woe is to me,” or, as the King James Version of the Bible has put it, “woe is unto me.”
Check out the following links to authors who have given more meticulous explanations: Jonathon Owen, Jan Freeman, and Arika Okrent (who points out that the same grammatical point persists in German and in Yiddish).