I think it can be translated as "in fact" when used as a correction or denial in the sense of "on the contrary". Here is the dictionary definition (which has no sign of "in fact" in the Italian > Engish translation but gives it as a valid translation in the English > Italian reverse translation) http://www.wordreference.com/iten/anzi and in the ReversoContext entry there are numerous examples of its use to mean "in fact" in the sense of a rebuttal or "to the contrary" http://context.reverso.net/traduction/italien-anglais/anzi
You'd only use "neither" if you're listing more than one thing. For example,"I am neither hungry nor tired." "Neither" also must go after "I am." The "I am" must stay together because you use the same verb for both "hungry" and "tired." Now, if you change the sentence to use two verbs, you would put neither after I, like this: "I neither am hungry nor need help." That way you are matching the verbs to each item in your list.