The Common Travel Area is only applicable for citizens of the Republic of Ireland, the Channel Islands, the Isle of Man, and the United Kingdom. It does not apply to any other citizens. A visa to enter the Republic of Ireland does not necessarily grant you the right to enter the UK, and vice versa.

IME the Irish Gardai tend to take the view that the best way of proving you are an Irish or UK citizen is by providing a passport stating that you are an Irish or UK citizen. Other documentation will also be accepted in line with the terms of the CTA, but you can expect more questions. I'm a British citizen born overseas so, to avoid the hassle, I take my passport when I go to Ireland. If you get your cancelled passport back, maybe take that just to be on the safe side. Better looking at it than for it.