Since Date implements Comparable, it has a compareTo method just like String does.

So your custom Comparator could look like this:

public class CustomComparator implements Comparator<MyObject> {
    @Override
    public int compare(MyObject o1, MyObject o2) {
        return o1.getStartDate().compareTo(o2.getStartDate());
    }
}

The compare() method must return an int, so you couldn't directly return a boolean like you were planning to anyway.

Your sorting code would be just about like you wrote:

Collections.sort(Database.arrayList, new CustomComparator());

A slightly shorter way to write all this, if you don't need to reuse your comparator, is to write it as an inline anonymous class:

Collections.sort(Database.arrayList, new Comparator<MyObject>() {
    @Override
    public int compare(MyObject o1, MyObject o2) {
        return o1.getStartDate().compareTo(o2.getStartDate());
    }
});

Since java-8

You can now write the last example in a shorter form by using a lambda expression for the Comparator:

Collections.sort(Database.arrayList, 
                        (o1, o2) -> o1.getStartDate().compareTo(o2.getStartDate()));

And List has a sort(Comparator) method, so you can shorten this even further:

Database.arrayList.sort((o1, o2) -> o1.getStartDate().compareTo(o2.getStartDate()));

This is such a common idiom that there's a built-in method to generate a Comparator for a class with a Comparable key:

Database.arrayList.sort(Comparator.comparing(MyObject::getStartDate));

All of these are equivalent forms.