The metric describing the surface of the unit sphere (with $x^1 = \theta$ and $x^2 = \phi$ ) is $$ [g_{ij}] = \begin{pmatrix} 1 & 0 \\ 0 & \sin^2{\theta} \end{pmatrix} $$ Find the covariant form of $ [A^i] = \begin{pmatrix} \pi \\ \pi / 4\end{pmatrix} $
Using $$ A_i = \sum_{j=1}^2 g_{ij}A^j $$ I get $ [A_i] = \begin{pmatrix} \pi \\ 0\end{pmatrix} $. Is this correct?
You apply the formula you've written down. The problem with your answer is that your are evaluating the metric tensor in $\pi$. What you should calculate is $$A_1 = \sum_j g_{1j} A^j = g_{11} A^1 = 1 \star \pi $$ and $$A_2 = \sum_j g_{2j} A^j = g_{22} A^2 = \sin^2 \theta \frac{\pi}{4} $$