Answer
a) $-1\;\rm \mu J$
b) $1\;\rm \mu J$
Work Step by Step
$$\color{blue}{\bf [a]}$$
We know that the mechanical energy is given by
$$E=U+K$$
This mechanical energy is constant and since the dipole oscillates between $\pm 60^\circ$, its kinetic energy at $\pm 60^\circ$ must be zero.
And hence, at $\pm 60^\circ$, its potential energy is given by
$$U_{\rm dipole}=-pE\cos\phi=-0.5 pE$$
where $pE$, from the given graph, is the maximum potential energy of the dipole, 2 $\rm \mu J$.
$$U_{\rm dipole} =-0.5(2)=-1\;\rm \mu J$$
$$E_{\rm at \pm 60}=-1+0=\color{red}{\bf-1.0}\; \mu J$$
$$\color{blue}{\bf [b]}$$
We know that the mechanical energy is conserved, so
$$K+U=-1\;\rm \mu J$$
when it the dipole aligned with $\vec E$, $U=-pE\cos0^\circ=-pE=-2\;\rm \mu J$
$$K-2=-1 $$
Hence
$$K=\color{red}{\bf 1.0}\;\rm \mu J$$
