This is something I've been wondering for a bit. At first glance it seems like it would be more animalistic, but after thinking a bit more I thought it would probably be pretty similar to what we do now (for example Kate Fox's SIRC guide to flirting or the way we use body language).
EDIT: I formulated this a bit weird and made a spelling error (please don't judge), sorry. I'm meant how the courting/mating ritual was for the Homo Sapiens in the neolithic and pre-neolithic period.
This is a question for /r/AskAnthropology. I'm an archaeologist who hangs out here with the historians because sometimes I find questions related to my field here instead of over there.
Homo sapiens are modern humans... I see you are trying to ask about the Neolithic and Paleolithic ("pre-neolithic"), so it would be better to ask, "What were human mating rituals like in the Neolithic and Paleolithic?"
I do not know the guide to flirting you mention, but you are most likely right that human mating rituals have not changed much over the millennia, but really it's hard to say for sure.
Think about what courting someone is like, what is the physical evidence? Maybe someone gives someone else flowers, which are then thrown out and rot in the trash, becoming indistinguishable from the flowers you gave your mom for her birthday or the flowers you bought to brighten up your kitchen in the winter. Maybe jewelry is exchanged, and then worn until death when it becomes indistinguishable from the other jewelry the person is wearing. Writing was invented after the Stone Ages, so poems and sonnets weren't being exchanged (the sonnet didn't exist for many thousands of years).
The point I'm trying to make is that certain human interactions leave very little evidence in the archaeological record. The archaeological record (the objects, bones and buildings left behind and preserved) is how we interpret the past. It is very difficult to guess at actions and thoughts based on a series of objects. We can see conflict by finding arrowheads lodged in hipbones, we can see religion or spiritual belief (we think) in artistic drawings on stone and burial practices, we can understand what people ate by analyzing bones, hearths and teeth. But we can't know what songs a Paleolithic mother sand to her child, or how a Neolithic teenager won the hand of the love of his/her life in marriage.
First of all, we are Homo sapiens. Brain size has not changed in a meaningful way in the past 200,000 years. Second, your answer question simply can't be answered because you are asking about a period with no written records about a conduct that does not get preserved in any other way. We can only speculate that it would be as varied as it is today.