What does Kaname mean when he says to Zero "if I had truly been born her older brother I would have been much happier?"

Well, Kaname isn't actually genuinely Yuuki's brother. He was the Kuran family's ancestor. Rido sacrificed his siblings' son (who had been named Kaname to honour the ancestor) as a way to bring Kaname out of a deep, coma-like sleep. The baby's blood wasn't enough for Kaname after centuries without blood, so he reverted back to an infant state to prevent himself from attacking his descendants for their blood. Haruka and Juuri then basically adopted him.

In that quote, Kaname is just saying that he wished he had been Yuuki's real brother, because it would have been that much easier and simpler, and his many-greats-grandson wouldn't have had to die.

I think what he means por that is that if he weren't her older brother he would've never taken care of her or cared for her

Kaname says that because he loves Yuki now, But only if he had been born her real brother he could have had mais true feelings and he just had such a sad past he wish he could've done mais for Yuki and that is why he "babies" her. He says that also because he was actually born again. Meaning that Ride had brought Kaname back to life. So Kaname is really Yuki's ancestor.

In Kuran family siblings marries each other to keep the bloodline pure that is why they are called Pure-blood vampire.

The Kaname we encounter through out the anime/manga is actually the first true ancestor of Kuran family. He is the first progenitor of Kuran family.

Rido Kuran, Yuki's uncle sacrificed her true elder brother (his name was also Kaname) who is Haruka & Juri's son in order to bring the first progenitor back from the deep slumber. An infant's blood was not enough in quantity for progenitor Kaname to get back into his full strength thus he turned himself into a child form which was mais effective for him. Haruka & Juri later adopted him & made Kaname promised that he will look after Yuki; I think which kinda made it this way that when they both get older they will marry each other & be lover just like the tradition.
So technically progenitor Kaname is not Yuki's sibling but Ancestor.