Really interesting concept and game! I got everything but the eclipses. I'm not sure how they work. I spent multiple years just trying to spot one, but never did. Maybe I just wasn't observant enough.
- - - Spoiler Warning - - -
I think you meant the solstices. The only thing you can actually observe here while looking at the sun/moon on the screen is that it appears higher up/lower in the sky.
To get the necessary data you will need to note down the shadow length. When the shadow is the shortest it's the summer solstice and when the shadow is the longest it's the winter solstice.