I'm not even sure if Hannukah (as a holiday) outdates Christmas. I'm not sure if Hannukah was celebrated in Hasmonean/Roman Judea.
Hanukah dates to at least the first century. Quoth wikipedia:
The ancient Jewish Historian Flavius Josephus narrates in his book Jewish Antiquities XII, how the victorious Judas Maccabeus ordered lavish yearly eight-day festivities after rededicating the Temple in Jerusalem that had been profaned by Antiochus IV Epiphanes. Josephus does not say the festival was called Hannukkah but rather the "Festival of Lights":
"Now Judas celebrated the festival of the restoration of the sacrifices of the temple for eight days, and omitted no sort of pleasures thereon; but he feasted them upon very rich and splendid sacrifices; and he honored God, and delighted them by hymns and psalms. Nay, they were so very glad at the revival of their customs, when, after a long time of intermission, they unexpectedly had regained the freedom of their worship, that they made it a law for their posterity, that they should keep a festival, on account of the restoration of their temple worship, for eight days. And from that time to this we celebrate this festival, and call it Lights. I suppose the reason was, because this liberty beyond our hopes appeared to us; and that thence was the name given to that festival. Judas also rebuilt the walls round about the city, and reared towers of great height against the incursions of enemies, and set guards therein. He also fortified the city Bethsura, that it might serve as a citadel against any distresses that might come from our enemies."
Records of Christmas only pop up later. Again quoting wikipedia:
The earliest known reference to the date of the nativity as December 25 is found in the Chronography of 354, an illuminated manuscript compiled in Rome.[