As you may know, Elrond and his brother, Elros, were half-elven, the sons of Eärendil and Elwing. The Valar gave them (along with the other living half-elves) the choice to follow either the path of the Elves or the path of Man. Elrond chose his elven side, remaining immortal, and Elros chose the Gift of Men and becoming the first King of Númenor. His line did not have a choice and were all thereafter mortal men, though with a very long life span (and some of his descendants, especially in later years, begrudged him that choice). Elrond's children were immortal until Elrond sailed west to Valinor, at which point they were given the choice to follow him there or remain in Middle Earth and become mortal. Arwen chose to remain to marry Aragorn. Thereafter, both being mortal, their child, Eldarion, was also counted among men. From the LotR wiki:
[Arwen] exercised this option [remain and become mortal], marrying Aragorn II Elessar, king of the Reunited Kingdom, and finally dying at the age of 2,901 years. Their son Eldarion and their daughters were not counted as Half-elven, but rather as Dúnedain restored.
So we know he's mortal, but not precisely how long he'll survive. I would wager, because of the renewed elven blood in his veins, around the same 500 years Elros's line started out at.