Continuity mistake: During the scene where he is first at the cave, look in the sky. First, there aren't any clouds then there are in the next scene. If you keep watching, the clouds vary from the different camera angles.
Other mistake: In the cave we see Tom Hanks drawing an analemma without a precise timekeeper, which is needed to draw one. What he could have done is a meridian rect, which is build with the meridian time. The analemma is the differential chart between the solar time and civil time.
Other mistake: When Tom Hanks throws the spear at the fish it's sticking straight up. He's too close for that to happen. He would've had to throw it straight up a long way for that to happen.
Continuity mistake: When Tom Hanks sees the ship from the beach, it is obvious there is a pane in front of the camera. However, as he is jumping up and down through the scene, his shadow changes angles from in front of him, to a very low angle behind him.
Visible crew/equipment: When he collects coconuts for the first time if you look in the lower right corner you can see some sort of light flashing, I'm guessing maybe a light meter or possibly a two-way radio that the crew was using.
Suggested correction: No, what we see is Tom Hanks drawing a chalk outline on a wall to approximate the date. There is no reason to assume he got it exactly right or knew exactly what he was doing. He probably started just by drawing lines on the cave wall to mark the days then noticed the light.
Greg Dwyer
I'm afraid "just drawing lines" doesn't create the well-crafted analemma you see in the movie. A very precise time-keeping device is necessary: just a minute off will dramatically change the shape.