Interested in Language
I think both (1) and (2) are fine.
(1) I don't think Bill will arrive until 8:00, do you?
(2) I don't think Bill will arrive until 8:00, will he?
Do you agree?
Thank you in advance
"Bill won't arrive until 8:00, will he?"
In the first sentence you are indicating that it is your idea and you want to know if your listener agrees. Hence, "I don't think.....Do you think?"
In the second sentence (as I have rewritten it) you are stating a hypothesis and are asking if it is correct. "
"Bill won't arrive.....Isn't that [statement] correct?"
In practice you could get away with either of your two statements, but there is a difference in nuance.
I hope that's not too technical. It is a minor point, to be sure.