Cosmology is the study of the world in relation to the universe and space. The first person to use the term cosmology was Thomas Blount. The first cosmologist is Aristotle because of his strong work related to space and universe. However, Aristotle is known for many things other than cosmology. His love of Physics helped him become the first cosmologist. He studied the Earth, water, air and fire as well as the spheres and stars.
He also studied the planets. He learned that the earth is cold and dry and its motion goes down. Its state of matter is solid. Aristotle probably learned some from his father Nichodemus due to his love of science. There are other famous cosmologists other than Aristotle.
The first cosmologist, in fact, would be the first individual to study and record physical events and features beyond the known environs. We don't know the first person to do this, and it's most likely that there were several such innovative thinkers gazing into space around the same period. This would be most likely after that time when all energy was needed just to survive.
Early sailors were cosmologists of a kind in that they used the stars as guidance for their journeys, thereby recognizing that there was some pattern and information beyond the Earth. Getting to the time when human events could be recorded, the ancient Greeks made use of navigational knowledge.
disregarding the Greek use of mythology of a way of understanding the universe, let's take Anaxagoras of Clazomenae as the first cosmologist for he claimed that the Moon shines only through the light it reflects from the sun, and that that lunar eclipses are a result of the earth blocking the sunlight in its path to the moon. He deserved a Nobel prize but was serious centuries ahead of his time.