The question should be unanswerable.
PART ONE, REASONS WHY THE QUESTION IS PROBABLY UNANSWERABLE
The universe should be about 13.787 billion - 13,787,000,000 - years old according to current evidence. Intelligent life has evolved on the planet Earth (possibly for the first time on Earth) about 4,600,000,000 years after the Sun and the Earth formed.
So it might have been theoretically possible for Intelligent life to evolve in our galaxy as early as 4,600,000,000 years after the formation of the universe, and so about 9,187,000,000 years ago - possibly much earlier if it is possible for advanced life to evolve much faster than it did on Earth.
Of course it was necessary for several generations of massive, short-lived stars to synthesize heavy elements in Supernova explosions and enrich the interstellar gas and dust with them in order for solid planets to form on which life could evolve, eventually become intelligent, and discover enough metals to build civilizations.
So it may have taken millions or even billions of years after the formation of the first stars and galaxies for stars and planets to form with enough heavy elements to eventually evolve intelligent, civilized beings. But I expect that the first civilized species in our universe could have evolved four billion years ago, or five billion years ago, or six billion years ago, or seven billion years ago, etc.
The star Kepler-444 is estimated to be about 11.2 billion years old, and it has four small rocky planets orbiting close to it. This implies that intelligent life might have evolved and developed civilization as long ago as about 6.6 billion years ago, which is about 2 billion years before the Sun "burned hot in space".
In "City on the Edge of Forever":
KIRK: Then what is it?
GUARDIAN: (The doughnut pulses bright in time with the words) A question. Since before your sun burned hot in space and before your race was born, I have awaited a question.
KIRK: What are you?
GUARDIAN: I am the Guardian of Forever.
KIRK: Are you machine or being?
GUARDIAN: I am both and neither. I am my own beginning, my own ending.
SPOCK: I see no reason for answers to be couched in riddles.
If the Time Guardian has been waiting since before the species Homo sapiens evolved about 200,000 years ago, and since before the Sun first shone about 4,600,000,000 years ago, it has been an artifact and/or a being for at least 4,600,000,000 years.
Of course since it is a time travel artifact it might have been sent back in time to over 4,600,000,000 years ago and that is not proof that there actually was a civilization over 4,600,000,000 years ago.
The ruins of a large city on the time guardian's planet are only:
SPOCK: And of considerable age. On the order of ten thousand centuries old.
And a civilization existing a mere million years ago was insignificantly old compared to the age of the Guardian of Time. But on the other hand there could have been civilizations on that planet much older than the ruined city that have no remaining traces for Spock to detect.
In TNG "The Chase" there is a puzzle left by an ancient species in the DNA of lifeforms on many different planets.
LAFORGE: This is not a natural design. Captain. This is part of an algorithm, coded at the molecular level.
PICARD: An algorithm? Are you saying that these DNA fragments are elements in some kind of computer programme?
LAFORGE: I know how it sounds, but there's no way this could be a random formation. This is definitely part of a programme.
CRUSHER: This fragment has been part of every DNA strand on Earth since life began there, and the other fragments are just as old. Someone must have written this programme over four billion years ago.
PICARD: So, four billion years ago someone scattered this genetic material into the primordial soup of at least nineteen different planets across the galaxy?
DATA: The genetic information must have been incorporated into the earliest lifeforms on these planets, and then passed down through each generation.
And the program creates a hologram with a message from a species that evolved over 4,000,000,000 years ago and found no other intelligent life in their part of the galaxy. So they seeded their genes on many young planets so that someday intelligent beings like them would evolve on those planets.
In the TAS episode "Beyond the Farthest Star" a strange alien starship is found orbiting a dead star:
KIRK: Can you identify that metal, or the design of the ship?
SPOCK: Negative to both, Captain. Unknown alloy, harder and lighter than any registered metal. It is not a recorded galactic starship design. Retro analysis of the ship's spectra dates it as having been in orbit here for slightly more than three hundred million years.
UHURA: It's beautiful. What kind of people could have built it, to touch even a starship with grace and beauty?
KIRK: A civilisation that advanced three hundred million years ago before life even emerged on Earth.
MCCOY: Barely an instant in eternity, Jim.
So that is an example of a civilization advanced enough for interstellar travel that existed about 300,000,000 years ago and is no more.
So species must have been evolving, building advanced civilizations, discovering warp drive and exploring the galaxy, and later dying out, for hundreds of millions of years.
And some former warp capable species may have retained their civilizations to the present but given up space travel, and other former warp capable species may have lost their civilizations but still exist as primitive hunter-gatherers.
And the question is a bit imprecise. Is it asking for the first humanoid warp capable species to evolve as a species, or for the first humanoid species to become warp capable?
My interpretation of the histories of several species in Star Trek, such as Vulcans and Bajorans, is that they developed advanced and possibly warp capable civilizations that fell and rose again a few times in their history, so that their present warp capable civilizations are not the first civilizations, or even the first warp capable civilizations, in the histories of those species.
And if that is the case for some species, should they be dated by the first or by the most recent date when they developed warp drive?
PART TWO, A POSSIBLY CORRECT ANSWER:
There is one possibly correct answer from one possibly canonical source describing the known to the Federation warp capable species considered the oldest at one specific time in Star Trek history.
in TAS, which has not always been considered totally canonical by all Star Trek creators and fans, the episode "The Jihad", which opens with a log:
Captain's log, stardate 5683.1. We're making a rendezvous with the Vedala asteroid on a unique mission. The Vedala are the oldest space faring race we know. They say something incredibly dangerous to the safety of our galaxy is developing and they have sent for selected specialists, including Spock and myself.
As Memory Alpha says:
The Vedala were a species associated with the Vedala asteroid. They were, as of the mid-23rd century, the oldest surviving spacefaring race known to the Federation. Physically, they had somewhat feline characteristics, including a tail, and a mighty roar.