How about this as a possibility?

Tá cumha orm i ndiaidh mo dhúchais
I long for my [home town/homeland/native place]

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

In that case, I'd suggest dúiche or dúthaigh, rather than dúchas:

Tá cumha orm i ndiaidh mo dhúiche.

or

Tá cumha orm i ndiaidh mo dhúthaigh.

I'd interpret Tá cumha orm i ndiaidh mo dhúchais more as "I long for my heritage".

Although it can refer to a place, dúchas is more the abstract concept of "heritage" and can even mean "(going back to) one's old ways or habits".

For a place, I think you'd normally clarify dúchas with a place noun baile dúchais "native home/home town", áit dhúchais "native place", tír dhúchais "native land", etc.

So another possibility is:

Tá cumha orm i ndiaidh mo bhaile dúchais. "I long for my native home."

Dúiche (and dúthaigh) specifically refers to such a place. I've also heard it in the diminutive, as dúthaighín:

Tá cumha orm i ndiaidh mo dhúichín/dhúthaighín.

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).