It is not easy to make a choice about which form of the future to use. One question to bear in mind is: is the activity planned, an intention or a spontaneous decision? The context will help you with this.
“Be about to” is a less common form of the future and means that the speaker is on the point of doing something. Using this definition, you can see that “construction is about to begin next month” is incorrect because it's not about to start.
In the example, “I am just about to go on a hiking trip to Nepal” the speaker is probably on the point of going on the hiking trip. However, I find it an unusual choice in this dialogue. I probably would have chosen another form of the future but I would need to see the context.
Again, this is the main point when choosing a form of the future: you need to take the context into account.