Aha. So the Irish does not actually contain a double negative then. Thanks.
From memory, I think that the DL dictionary hint listed it as not either but neither. Indeed, just looking up at the top of this page again, I see that the official translation is "I have neither". That has exactly the same meaning as "I don't have either" but is grammatically different.
The simple answer is Ná mise, which is essentially "Nor me" or "Me neither".
The more complicated answer is that "Neither have I" is ambiguous in English.
"John hasn't been to Spain" - "Neither have I"
"Deirdre hasn't eaten breakfast" - "Neither have I"
"Deirdre hasn't been eating breakfast" - "Neither have I"
"This guy doesn't have any money" - " "Neither have I"
The various perfect tenses in English are often just translated as the past tense in Irish, but you might have to restructure the sentence entirely in some cases. And you would usually repeat the actual verb in Irish (Níor ith mé ach oiread), not just the auxiliary "have". In the case of the possessive "have", it's a bit complicated by the the fact that "I" isn't the subject of Tá airgead agam, so you would have to say something like Níl airgead ar bith agam ach oiread If Ná mise wasn't enough.