I don't think your Estonian ID card will be sufficient for travel to Ireland.
First, while your ID card does state that you are a national of Ireland, it is not authoritative, because Estonia does not have the authority to declare that you are an Irish citizen. Only Ireland has that authority.
Second, Timatic, the database which airlines use to verify travel documents, does not list this ID card as sufficient for Irish citizens to enter Ireland. Timatic currently states with regard to Irish citizens:
- Passports and other documents accepted for entry issued to nationals of Ireland (Rep.) must be valid on arrival.
- Nationals of Ireland (Rep.) with a passport card.
- Passengers with a proof of nationality issued to nationals of Ireland (Rep.) and British subjects for travel between Ireland (Rep.) and Great Britain and Northern Ireland only.
As soon as the airline enters in that you are an Irish citizen, this will come up, and the Estonian ID card is not listed here.
Critically, Timatic says the same thing for the return journey, regarding the passport you must carry.
If you haven't got your passport or passport card, you should obtain one. You can apply for it online and have it delivered to you (or to the embassy) in Estonia.