Arachne really 'isn't' in the comic books in that she's a character. She's MENTIONED on the first page of Ultimate Spider-man #1. I'm looking at the thing right now. It's just a spider that Norman Osborn is holding and he's saying,
"THE STORY GOES THAT ATHENA- YOU KNOW ATHENA, RIGHT? SEEMS SHE HEARD THERE WAS THIS WOMAN ON EARTH- A MERE MORTA, LIKE YOU AND ME- WHO HAPPENED TO BE A BETTER SPINSTRESS THAN SHE WAS. ATHENA WASN'T TOO HAPPY TO HEAR THIS AND SHE CAME DOWN TO EARTH AND DESTROYED THE WOMAN'S CREATIONS. WHEN THIS MORTAL GIRL SAW WHAT HAD HAPPENED- THAT SHE HAD INSULTED THE GODS AND THAT HER LIFE'S WORK HAD BEEN DESTROYED- SHE HANGED HERSELF.ATHENA TOOK PITY ON THIS POOR GIRL, AND TOUCHED HER ON THE FOREHEAD WITH A MAGIC LIQUID AND SAID: "YOU SHALL NOT DIE, ARACHNE. INSTEAD YOU SHALL BE TRANSFORMED AND WEAVE YOUR WEB FOREVER."
So, she's not the character Taymor created. She's merely referenced in her mythic self. If you want to read into it and say the spider Norman is petting is Arachne, fine. But she doesn't talk or sing or want to have sex with Peter (who in the Ultimates is like 15, so ew).
It would be an interesting legal decision that could be made on whether Taymor really 'owns' the characters of Arachne and the Furies. Personally, I'd love to see it play out so we could see!
"Are you sorry for civilization? I am sorry for it too." ~Coast of Utopia: Shipwreck