You are only technically correct. Eire is the Irish name for Ireland, but is not commonly used in English. In Ireland, everyone calls Ireland Ireland. Similarly, the language is Irish, not Gaelic. You only hear the words "Eire" and "Gaelic" used outside Ireland, typically in America (except for the small section of the population who actually speak Irish).

If you were to use the Irish term Eire, then it would make sense to use the Irish name for Dublin, which is Baile Atha Cliath, or Dubh Linn, depending.