All you had to do was measure when it was local Noon, i.e. when the Sun was at the Southernmost position.
You could either measure it with a compass or with a gnomon. The gnomon is basically a pole that projects a shadow. Keep marking where the shadow moves during the day, and the apex of the parabolic curve that you have drawn shows your local Noon.
Once that you have your local noon, you can set your clock and measure by how much it deviates until the following Noon.
You can find-tune the clock until it keeps time reliably between two noons.
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