How to derive an expression for the optimal importance distribution?
Hint

Well, it looks like the question tells you what the optimal importance distribution is! It's $p(\mathbf x_k | \mathbf x_{k-1},\mathbf y_{1:k})$. So the problem boils down to computing a closed form expression for this conditional density using the given state space model.

More Hints

The transition model tells you $p(\mathbf x_k | \mathbf x_{k-1})$ is a multivariate Gaussian. The observation model tells you $p(y_k|\mathbf x_{k})$ is another multivariate Gaussian.


Can you write these multivariate Gaussians in closed form?
Can you express $p(\mathbf x_k | \mathbf x_{k-1},\mathbf y_{1:k})$ in terms of $p(\mathbf x_k | \mathbf x_{k-1})$ and $p(y_k|\mathbf x_{k})$?


Almost complete solution

\begin{eqnarray}
p(\mathbf x_k | \mathbf x_{k-1}, \mathbf y_{1:k}) &amp;=&amp; p(\mathbf x_k | \mathbf x_{k-1}, \mathbf y_{1:k-1}, y_k) \label{1}\tag{1} \\
&amp;=&amp; p(\mathbf x_k|\mathbf x_{k-1}, y_k) \label{2} \tag{2}\\
&amp;=&amp; \frac{p(y_k|\mathbf x_k,\mathbf x_{k-1}) p(\mathbf x_k | \mathbf x_{k-1})}{p(y_k | \mathbf x_{k-1})} \label{3} \tag{3} \\
&amp;=&amp; \frac{p(y_k|\mathbf x_k)p(\mathbf x_k | \mathbf x_{k-1})}{p(y_k | \mathbf x_{k-1})} \label{4}\tag{4}
\end{eqnarray}
where (\ref{1}) is just another way of writing $\mathbf y_{1:k}$, (\ref{2}) follows from Markov property (Eq. 4.2 of the book), (\ref{3}) follows from Bayes' theorem and (\ref{4}) follows from conditional independence of measurements (Eq. 4.3 of the book).

Using the model given in Eq. (7.49) in the original question we have 
$$
p(y_k|\mathbf x_k) \sim  N((1 \;\; 0) \mathbf x_k,10^2) \label{5}\tag{5}
$$
and
$$
p(\mathbf x_{k}|\mathbf{x}_{k-1}) = N\left(\left(\begin{array}{}
1 &amp; 1\\
0 &amp; 1
\end{array} \right)\mathbf x_{k-1}, \left(\begin{array}{}
1/10^2 &amp; 0\\
0 &amp; 1
\end{array} \right)\right). \label{6}\tag{6}
$$
To compute $p(y_k | \mathbf x_{k-1})$ we express $y_k$ in terms of $\mathbf x_{k-1}$ as follows:
\begin{eqnarray}
y_k &amp;=&amp; (1\;\;0)\mathbf x_{k} + r_k \\
  &amp;=&amp; (1\;\;0)
\left(\begin{array}{}
1 &amp; 1\\
0 &amp; 1
\end{array} \right)\mathbf x_{k-1} + (1\;\;0)\mathbf q_{k-1} + r_k \\
&amp;=&amp; (1 \;\; 1) \mathbf x_{k-1} + (1\;\;0) \mathbf q_{k-1} + r_k
\end{eqnarray}
which implies that
$$
p(y_k | \mathbf x_{k-1}) \sim N\left((1 \;\; 1) \mathbf x_{k-1}, \frac{1}{10^2}+10^2\right).\label{7}\tag{7}
$$
Finally you should use Wikipedia or Eq. (A.1) from Appendix A.1 of the book for the expression for a multivariate Gaussian density function. 

Time for grunge work. Plug (\ref{5}), (\ref{6}) and (\ref{7}) into (\ref{4}) and finish with some algebra.