In Aristotle's Physics the existence of time is deeply called into question and reinterpreted in an apparently phenomenological manner. I think the conception of time in Kant and Heidegger can benefit from comparison to Aristotle. I will carry out this comparison in three parts.
For Aristotle the beginning and end of any time is ideal and can be described like temporal geometric points (i.e., they have no size). Between these points is an indeterminate duration. Time only exists in this structure:
beginning -------------- end
This does not mean that for Aristotle time is composed out of units of beginnings and ends, or even overlapping beginnings and ends. Instead, the experience of time always structurally contains these moments.
When we become aware of something it is always awareness of a difference - a change has taken place. This awareness is the awareness of an end, and the prior state before the change is the start.
Imagine that someone throws a ball (or throw a ball into the air yourself). You see it in its trajectory over numerous points in space. Each of those points is an awareness of change that takes its reference to a beginning. Here time is not a duration but a certain mode of awareness about things. In order to have a duration one must measure time by something. We are not concerned with measuring time, but I can remark that such measuring is done traditionally through movements (perceived to be) of a fixed kind (e.g., the movements of planets).
In this experience of time, the beginning doesn't come before the end, but both come together in awareness. From this, Aristotle is able to make the point that there is no such thing as the beginning of movement. One can only encounter movement in its continuing to move or stopping. (While I am sticking to movement I should at least point out that all change of characteristics are understood in this way by Aristotle.) One can verify this with reference to ones own experience of change.
Now, the structure of time with a beginning an indistinct duration and an end is something that is all at once for awareness, but looks like a line drawn out (as we map it to space), and so it is easy to think that we can point the the beginning. In this regard, of course, there is a beginning, but if we want to speak of a beginning in time without an end, then we are not speaking about our primordial experience of time but of time converted already into a duration, and so we mean the time.
The next part (Part II) will concern Kant and how the indistinct-duration (that Aristotle recognized) in the experience of time is given priority. This indistinct measure is not broken up moment to moment, but considered as infinite. Then pure concept (category) of cause is used to provide the same model of beginning, indefinite duration, and end that is found in Aristotle.