Unique carbons on rings
Yes, one would try to find points or lines of symmetry in general to check for equivalent ring atoms. It immensely helps to have a model and a lot of practice.

In your specific example, notice that you have two lines of symmetry in your (idealized) diagram: one passing through the sp3 carbons, and one crossing the two double bonds. You can thus "fold" the diagram into "quarters", leaving you with two non-equivalent atoms.