"naik" means go up, "turun" go down, so "Saya tidak mau naik/turun" is sufficient. "ke atas", "ke bawah" are unnecessary.
Saya tidak mau naik = I do not want to go up
Saya tidak mau naik ke atas = I do not want to go up upward :D
maju ke depan
mundur ke belakang
create new ...
I've usually heard ...mau naik...(gunung, bukid -mountain, hill). And "atas" I've heard used to mean the floor above, or if your underwater and you want to go to the surface. So i feel like the English translation is missing the end part of the sentence.
i think you're right. the english does feel odd, like it has kind of stopped short. though i can see why it's been translated as it has. i just feel that we'd possibly say something more like, 'i don't want to go up there/ up above/ upstairs'.