As Jno, and others, indicate, all EU countries are required to recognise proof of EU citizenship (either through a passport or, for those countries that have them, a national ID card) right up until the expiry date of the relevant document. As long as you're travelling to Southern Cyprus you'll definitely have no problems.
The Turkish Republic of Northern Cyprus is still officially part of the EU but the administration is not recognised by any country other than Turkey. That administration might therefore choose to apply different rules to those which it's officially obliged to recognise.
Chris