Cirdan is the oldest named Elf at the end of Lord of the Rings, who takes part in events.
HoME 12 notes that his original name was "Nowe", which is an archaic form with an uncertain original meaning. It also discusses his pre-eminence as a ship-builder, even during the Teleri's stay on the shores of the Sea of Rhun during the Great March. These, taken together with his relationship to Elwe and Olwe, make it almost certain that he dates to Cuivienen, although whether he was one of the First or a later generation is uncertain.
The first three (male) Elves to awake were named Imin, Tata and Enel, who each awoke with their designated spouse (as did all of the First). By contrast, the leaders of the three clans of the Eldar were Ingwe, Finwe and Elwe.
Ingwe is probably not Imin
This is deduced from writings in HoME 10, where Indis of the Vanyar is noted as being Ingwe's sister (therefore they had parents and couldn't have been among the First). These weren't taken up in the published Silmarillion.
Finwe is not Tata
- Finwe married Miriel
- Miriel had a mother-name
- Therefore Miriel had a mother
- Therefore Miriel wasn't one of the First
- Therefore Miriel couldn't have been Tata's designated spouse
Elwe is not Enel
- Elwe had two brothers: Olwe and Elmo
- Therefore Elwe (and his brothers) had parents
- Also, Elwe married Melian
- Therefore Elwe didn't have a designated spouse
Finwe and Elwe are also not among the First (on the designated spouse criteria), but Ingwe may be if we reject Indis as his sister.
If we accept Indis as Ingwe's sister, then Ingwe is not one of the First even if we stretch the definition of "sister", as otherwise Indis would have also had her own designated spouse and therefore would not have been able to marry Finwe.
All of the first clan became Eldar, therefore Imin was either captured by Melkor or he left for Aman. Half of the second clan and about two-thirds of the third clan became Eldar, with the rest remaining as Avari. Tata and Enel may have become Eldar or Avari; it's not stated anywhere. Either way (and unless they were captured by Melkor - which is also possible), they're still around at the time of LotR; they just don't come into the stories.
It can be fairly assumed that Ingwe and Olwe still live at the end of LotR and would therefore also qualify, along with Cirdan, as being among the oldest living sentient beings. Ingwe was one of the three Elves who visited Valinor with Orome and returned to convince the Elves to go; Olwe was one of the two leaders (with his brother Elwe) of the Teleri on the march west (see the Silmarillion, Of the Coming of the Elves and the Captivity of Melkor). As of the end of the Silmarillion, both Ingwe and Olwe were the kings of their respective people in Valinor.
Galadriel is the only one left of the leaders of the Noldor who took part in the rebellion (Letter 297). Glorfindel and Gildor Inglorion are of course also noted as Noldor. There are variant stories about the history of Celeborn, but published writing notes him as a kinsman of Thingol who lived in Doriath. He may or may not date back to the Great March or even Cuivienen. Thranduil also goes back to Doriath, and Legolas may even be of similar vintage (he does immediately recognise the Balrog, and Tolkien notes in Letter 144 that "it is observable that only the Elf knows what the thing is" - thin evidence but evidence nonetheless).
Among the Dwarves the various reincarnations of Durin should be considered, although they don't directly take part in events of LotR. However, if Dwarf myths are true, one could argue that Durin was originally incarnated (by Aule, then by Iluvatar) before the Elves.
Treebeard claims to be one of the first Ents that were awoken by the Elves:
Only three remain of the first Ents that walked in the woods before the Darkness: only myself, Fangorn, and Finglas and Fladrif (Two Towers)
This makes him quite old indeed, comparable in age to Cirdan at least. However, he also notes that there are other trees around that are even older than he is:
But there are hollow dales in this land where the Darkness has never been lifted, and the trees are older than I am. (Two Towers)
Finally, when discussing Treebeard, it's always important to note Tolkien's warning in Letter 153:
Treebeard is a character in my story, not me; and though he has a great memory and some earthy wisdom, he is not one of the Wise, and there is quite a lot he does not know or understand.
Gwaihir and Landroval date to the First Age; the 1937 Silmarillion names them as two of the Eagles that rescued Beren and Luthien from Angband, but Christopher Tolkien suppressed the names in the published Silmarillion based on evidence he has since rejected. This is all discussed in HoME 5.
There's no indication as to whether or not Thorondor is still alive. He may be the Lord of the Eagles in the Hobbit; we know that the Lord is not Gwaihir, because Gwaihir had only carried Gandalf twice before the destruction of the Ring ("Twice you have borne me, Gwaihir my friend" - RotK - from Isengard and Zirak-zigil), yet the Lord had carried Gandalf in the Hobbit.
Ungoliant is old, but probably not as old as another answer indicates. Let's have the Silmarillion quote again, but with different emphasis:
The Eldar knew not whence she came; but some have said that in ages long before she descended from the darkness that lies about Arda...
It should be obvious that this story of her origin is therefore a legend of the Eldar, and not an authorial statement by Tolkien. Her most likely origin is a corrupted nature spirit, as outlined in Of Aule and Yavanna:
When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar and the olvar, and some will dwell therein, and be held in reverence, and their just anger shall be feared.
Shelob dates to at least early Second Age: "but still she was there, who was there before Sauron, and before the first stone of Barad-dur" (Two Towers) - with "there" being Mordor, and Barad-dur being begun in SA 1000 (RotK Appendix B).
There was Eru, the One, who in Arda is called Ilúvatar; and he made first the Ainur, the Holy Ones, that were the offspring of his thought, and they were with him before aught else was made.
And you can throw Bombadil into that wherever best suits your own favoured theory as to who or what he is.