Antipodal mapping $f:S^n\to S^1$ I know that " There does not exist an antipodal mapping $f:S^{n}\to S^{n-1}$" by Borsuk-Ulam. But, ¿There does not exist an antipodal mapping $f:S^{n}\to S^{1}$ (continous)?

No, embed $S^1 \hookrightarrow S^{m-1}$ equatorially so that $(x_1,x_2) \mapsto (x_1, x_2,0,\dots,0) \in S^{m-1}$ and note that such a map would provide an odd map $S^n \to S^{m-1}$.

I guess if you wanted to do this from scratch:

Take $f:S^n \to S^1$ an odd map. So, we can embed $i:S^1 \hookrightarrow S^n$ equatorially, and note that the composition $f \circ i:S^1 \to S^1$ is nontrivial since its degree is $\pm 1$, while it factors through $S^n$ which has trivial $\pi_1$.