Of the named Elves it's almost certainly Cirdan, who was on the Great Journey and possibly even dates back to Cuivienen. read more
The six oldest elves, in order, are; Imin Tata Enel Iminyë Tatië Enelyë The first three are male, the second three female. Tolkien never said what happened to them afterwards. We can assume that Imin and Iminyë went to Valinor along with the other Vanyar, since none of that kindred stayed behind. read more
If we assume that at least some did, and if they're still alive, then they would be the oldest Elves in Middle-earth. Edit to Add: Some dates from the Annals of Aman (HoME 10), all given in YT (Years of the Trees; 1 YT = approx 10 Years of the Sun): 1050: The Awakening of the Elves. 1085: Orome finds the Elves. 1090: The Battle of the Powers begins. read more