Learn English – Is it possible to use “after a long time” or “after a long interval” this way

I don't believe that "for many times" is grammatical in any variety of English.

^{(Well, unless you contrive an example where "times" is the plural of "a given time of occurrence" and there's an elided "of the": "Class times are listed below. For many times, you can click the link for more information.")}

You can use "for" with an ordinal: for the first time, for the 20th time, for the last time. Notice that it's always time, not times, in these examples. You can also use "for" with a duration: for 100 years, for the summer, for a long time.

"Many times" behaves the same way as "100 times" or "just a few times", and adding a "for" is not grammatical.

Mark has been to Hong Kong many times.
Mark has been in Hong Kong for two years now.

After Jessica failed 10000 times, she finally created a working robot.
After Jessica failed for the 10000th time, she tried a different approach.

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.