I am given the following discrete system
Now from what I understand, since
therefor the system is causal as the output is depending on the past values of the input. Right?
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You are correct that $y[n]$ only depends on past input values for $n \gt 0$, but when $n \lt 0$, that is not the case --- then $y[n]$ is dependent on future input values, regardless of what $x$ is.
To see this: $$ y[-2] = x[-(-2)] = x $$ so the output at time $n=-2$ is dependent on the input at time $n=2$.