Whether you use existing classes for dates and times or develop your own, the Schedule class you're designing will have to contain events with names, start and stop dates and times, and the methods needed to do the checking you've described. At a minimum, I suggest you need an Event class (or the name of your choice) to describe the events which will make up the items in the schedule.
If you proceed down the events as ints path, how will you name the items? How will you determine start and stop dates and times. For the latter, you could count the number of minutes from some point in time (your birthday?) and then calculate the date and time of day from that, but that's where using the existing classes begins to look a lot more attractive.
It's not an easy problem, and I'm not sure you've thought it through very far. Give your design some more thought and determine how you'd answer some of the questions I've raised. Others will probably occur to you as you proceed.