Let N be a prime integer. We know that the element $c=(0)-(\infty)$ generates the torsion subgroup of $J_0(N)$ and it has order Num( (N-1)/12). Now, there is a natural map $\pi^*:J_0(N) \rightarrow J_1(N)$, coming from the covering map $\pi:X_1(N) \rightarrow X_0(N)$. My question is what is the image of c under this map? Specifically, is it possible for $\pi^*(c)=0$?
The fact you mention about $(0) - (\infty)$ generating the torsion subgroup of $J_0(N)$ is Theorem 1 (Ogg's conjecture) on the first page of Mazur's paper "The Eisenstein Ideal". I recommend you actually read this paper. If you get as far as page 2, you will find a "Theorem 2 (twisted Ogg's conjecture)" which concerns the Shimura subgroup $\Sigma$. The construction of this subgroup essentially identifies it with the kernel of $J_0(N) \rightarrow J_1(N)$, and Proposition 11.6 of ibid. shows that $\Sigma$ is of multiplicative type, so BCnrd's remarks apply to the general case.
I don't think pi^*(c) can be 0. Suppose pi^*(c) = div(f); then f would be a map from X_1(N) to P^1 of degree about N. But in fact any such map is of degree at least ~N^2, i.e. the gonality of X_1(N) is bounded below by a constant multiple of N^2. This was proved independently by Zograf ("Small eigenvalues of automorphic Laplacians in spaces of cusp forms") and Abramovich ("A linear lower bound on the gonality of modular curves.")
Update: As Kevin points out in comments I should say "for N large enough." But "large enough" is effective here since the constants in the gonality bounds are effective (albeit small.)