How could you find the preimage of an isogeny function? How do you know if an isogeny is surjective or not, and how do you tell how many points on E maps to E'? Does the answer lie in the degree of the isogeny function?
Isogenies are always surjective, but there's a nuance. They are surjective over the algebraic closure. The correct statement would be "for every $\mathbb{F}_{17}$-point on the green curve there are three $\bar{\mathbb{F}}_{17}$-points on the blue curve which map to it".

To find the preimages of a point, write down a polynomial system and find its solutions. A computer algebra system (e.g., SageMath) may help. The number of solutions is obviously related to the degree of the polynomial system.