In the Sillmarillion, there were a huge number of creatures who participated in "The Music of the Ainur," which foreshadowed the creation and history of Earth.
Many of these entered Arda and dwelt there when it was created. The 15 most powerful were the Valar, the single most powerful of whom was Melkor, the original Dark Lord, of whom Sauron was only a servant.
I would assume Tom B. was one of these Ainur, who were called the Maiar. They were mostly the followers and people of the Valar, but Tom was probably a free-lancer. If so, he was older than Middle Earth or Arda itself. Sauron, Saruman and Gandalf were other Maiar who were presumably older than Arda. Although some Maiar reproduced, so some of them might not be older than the Earth.
Treebeard was born(?) on Middle Earth and at the end of the Third Age might be the oldest "native" still alive.
I thought that the Maiar were beings on the same "level" as Gandalf, Saruman, and Sauron all of whom were affected by the ring. Bombadil seems to have powers greater than these (or at least greater than Gandalf and Saruman.)
Tolkien gives a comprehensive account of the great deeds and events of the three ages. He does not fully preclude the possibility of other entities arising outside the history he provides. The origin of the ancient Ents, for example, remains a mystery. Where and from what Tom Bombadil came I believe was meant to remain a mystery for ever ("forever" outside Tolkien-speak), adding to the mystery and depth of the world Tolkien creates.
Eventually, it would all become boring if Tolkien's written history covered all that transpired in the world he creates.