The oldest character in the Lord of the Rings is either Saruman or Sauron. They are both Maiar, Ainur who entered Aman at the very beginning of time. They're essentially demi-gods. They are similar to the Valar, the Ainur who participated in the creation of the universe, but they're likely less powerful.
Note that I don't include Gandalf as a possibility, since the Silmarillion makes clear that Curunir (Saruman) was the eldest of the Istari (wizards):
Of these Curunir was the eldest and came first, and after him came Mithrandir and Radagast, and others of the Istari who went into the east of Middle-earth, and do not come into these tales.