= Solution
Evaluation is surjective as a sheaf morphism: near $p$, any prescribed value is supplied by a constant <holomorphic function>. We therefore have a <short exact sequence of sheaves>
$$
0\longrightarrow\mathcal I_p\longrightarrow\mathcal O_X\xrightarrow{\operatorname{ev}_p}\mathbb C_p\longrightarrow0,
$$
where $\mathcal I_p$ is the <point ideal sheaf> and $\mathcal O_X$ the <sheaf of holomorphic functions>. By part (c), the <skyscraper sheaf> is flasque, so $H^0(X,\mathbb C_p)=\mathbb C$ and $H^i(X,\mathbb C_p)=0$ for $i>0$.
On the connected compact <elliptic curve>, the <maximum modulus principle> makes every global <holomorphic function> constant. Consequently $H^0(X,\mathcal O_X)=\mathbb C$, and evaluation on <global sections> is an isomorphism. Also $H^1(X,\mathcal O_X)\cong\mathbb C$: by <Serre duality for compact complex manifolds>, its dual is $H^0(X,K_X)$, generated by the nowhere-zero form $dz$ descended from $\mathbb C$. Higher cohomology of $\mathcal O_X$ vanishes by the <Dolbeault theorem> in complex dimension one.
The <long exact sequence in sheaf cohomology> thus begins
$$
0\to H^0(X,\mathcal I_p)\to\mathbb C\xrightarrow{\sim}\mathbb C
\to H^1(X,\mathcal I_p)\to\mathbb C\to0.
$$
It follows that
$$
\boxed{H^i(X,\mathcal I_p)\cong
\begin{cases}0,&i=0,\\\mathbb C,&i=1,\\0,&i\geq2.\end{cases}}
$$
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