Language and culture intertwines and it is not possible to accurately say which comes first, but Language seems to be necessary for a culture to develop. Culture means a way of life and this could not develop unless a group of first peoples had some means of communicating with each other. This then is my reason for saying language comes first.
And the word culture has so many meanings, unlike language which essentially means the sounds people make to convey meaning to others. As an example, if there were only two people on an island all by themselves and they met. Neither had known any other human being and one of the two knew where a good water hole is and where berries were plentiful. But not knowing how to get this information to the other, he invented a language of sorts.
a grunt to get attention along with gestures by hand to show how berries were picked off the bush and put in the mouth. After several repeats, the other understood. He was in need of nourishment and because of this lack, was not too keen on understanding what the other was 'talking about'. But he remembered the sounds mad by his friend that represented water and berries.
A far-fetched notion to prove language came first, but something like this could have happened, and who is around that know precisely how language began. However it is traced and studied, its origin eludes the best linguists. So htis 'tongue in cheek' scenario makes about as much sense as asking the question in the first place.