I can't think of any English term that is regularly used for this – at least, not in a noun form.
As Steve Ives mentioned in his answer, this is how I would ask the question:
What time are you leaving?
The term departure time is grammatically correct, but that expression is normally reserved for transportation: planes, trains, and busses. And it generally refers to the time that the conveyance departs (everyone knows that passengers must arrive earlier than that so they can all board before departure.)
If you want to tell your friends that you'll be leaving the campus at 5 o'clock, you'd say something like:
I should be leaving the campus around five.
or:
I should be done at the campus around five.
You'd only use departure time if you were trying to be funny by making something ordinary (i.e., leaving the campus) sound very official:
My departure time from campus will be 5 o'clock.
I could say a father saying this to his young children, if they were leaving for a trip the next day:
Our departure time will be 8 AM.
but that would be overly formal and therefore deliberately humorous. A more natural way to say it would be:
I want us to get out of here by eight.
or:
Let's be pulling out of the driveway before 8 AM.
The same holds for arrival time. We typically say something like:
We should be there a little before 10.
not:
We have a 9:55 estimated arrival time.
unless we are trying to be somewhat humorous by being overly technical.
If I may be so bold, I would say neither is correct.
Here is what I hear speakers say.
I have been waiting for her to be back for a long time.
I have been waiting for her to be back for quite a while.
In this context, quite and long have the same meaning and would be redundant next to each other.
I hope this helps.
Best Answer
There is a lot of confusion because of your example sentences, which seem to be drawing attention away from the issue you want to discuss.
Stay up is a phrasal verb that means stay awake or remain out of bed. If I say, "I stayed up until midnight last night," that essentially means "I went to bed at midnight last night," or, "I fell asleep at midnight last night."
So, the phrase stay up for work seems a little awkward. It's not something I would normally say, but I suppose I could contrive of a situation where I might say it. For example, let's say I know someone who is on call. They tell her, "You need to be ready to come into work if we call you. However, if we haven't called you by a certain time, you won't be called at all." If I don't know what that time is, and I'm curious, I could ask:
meaning, what time were you able to finally go to bed?
Perhaps you meant "stay at work", which makes a lot more sense:
meaning, "What time did you finish work last night, and head home?"
If I wanted to express with one of those two dastardly prepositions (by or until), I would use until:
Another possibility is that the person starts work at an odd time (say, 3AM). If I knew this person didn't go to sleep before starting work, I might ask:
Your sentence asks about until/by + what time, but many of these questions are asked with "how long" (or "how late") instead, which is why it's hard to give you a straight answer to your question.
Here's a situation where I might use this phrasing. You and I plan to go camping this weekend. I need to get some details before we can finalize our plans. Later tonight, I'll start doing some research. That is the backdrop for this conversation:
Until what time can I call? is asking how late it will be before it would be considered impolite to rouse you from bed with a phone call.