Here is the setup as I understand it. Let $y = (y_1, \ldots, y_n)$. The sampling distribution for $y$ can be expressed as
\begin{equation}
p(y|x) = \prod_{i=1}^n \textsf{N}(y_i|x,\sigma_i^2) .
\end{equation}
Once $y$ is observed, $p(y|x)$ becomes the likelihood for $x$. The prior distribution for $x$ is
\begin{equation}
p(x) = \textsf{N}(x|\mu,\sigma^2) .
\end{equation}
Given this setup, the posterior distribution for $x$ is given by Bayes' rule:
\begin{equation}
p(x|y) = \frac{p(y|x)\,p(x)}{p(y)} = \textsf{N}(x|m,s^2) ,
\end{equation}
where
\begin{align}
s^2 &= \left(\frac{1}{\sigma^2} + \sum_{i=1}^n \frac{1}{\sigma_i^2}\right)^{-1} \\
m &= s^2\left(\frac{\mu}{\sigma^2} + \sum_{i=1}^n \frac{y_i}{\sigma_i^2}\right) .
\end{align}
One may confirm the answer by checking that $p(y|x)\,p(x)/p(x|y)$ does not involve $x$ (after simplification).
The values for $m$ and $s^2$ may be found by using a property of the normal distribution. Given some $K > 0$, let
\begin{equation}
h(x) = \log\big(K\,\textsf{N}(x|m,s^2)\big) = -\frac{(x-m)^2}{2\,s^2} + C ,
\end{equation}
where $C$ does not involve $x$. We may obtain $m$ by solving $h'(x) = 0$ for $x$. In addition, $-1/h''(x) = s^2$.
We can apply this approach as follows. Let $K = p(y)$. Then $h(x) = \log\big(p(y|x)\,p(x)\big)$.
