= Solution
For a homogeneous <continuous-time Markov chain>, the <holding time> in a living state $r$ before the next transition has survival <probability> $e^{-\lambda_rt}$, where $\lambda_r=\sum_{s\ne r}q_{rs}=-q_{rr}$. Indeed, conditional on still occupying that state, the <probability> of leaving in the next $dt$ is $\lambda_rdt+o(dt)$, so its survival function satisfies $S'(t)=-\lambda_rS(t)$ and $S(0)=1$. Integrating gives the <mean holding time from a transition intensity matrix>:
$$
\boxed{\tau_r=\int_0^\infty e^{-\lambda_rt}\,dt=\frac1{\lambda_r}=-\frac1{q_{rr}}.}
$$
This is the mean time until any allowed exit from that state, not necessarily the patient's remaining lifetime.
For confidence limits, use the supplied <standard error> $s_r$ of the corresponding diagonal generator estimate. Changing its sign does not change its <standard error>, so it is also the <standard error> of $\widehat\lambda_r$. The <delta method>, with derivative $d(1/\lambda)/d\lambda=-1/\lambda^2$, gives
$$
\operatorname{se}(\widehat\tau_r)\approx\frac{s_r}{\widehat\lambda_r^2},\qquad
\widehat\tau_r\pm1.96\frac{s_r}{\widehat\lambda_r^2}.
$$
The resulting approximate 95% <Wald confidence intervals> are
$$
\boxed{\begin{array}{c|r|r|c}
\text{state}&\widehat\tau_r\ (\text{years})&\operatorname{se}(\widehat\tau_r)&95\%\text{ interval}\rule{0pt}{2.5ex}\\\hline
1&11.765&0.498&(10.788,12.741)\\
2&5.525&0.644&(4.263,6.787)\\
3&2.710&0.181&(2.356,3.064)\\
4&1.949&0.258&(1.444,2.454)
\end{array}}
$$
The diagonal <standard errors> already incorporate the <covariance> among outgoing rates. They should not be replaced by the square root of a sum of squared off-diagonal <standard errors>, which would silently assume those rate estimates independent.
Alternatively construct a positive rate <confidence interval> and apply the <confidence interval for an inverse rate> transformation. Here every normal rate interval has a positive lower endpoint, giving
$$
\left[\frac1{\widehat\lambda_r+1.96s_r},\frac1{\widehat\lambda_r-1.96s_r}\right].
$$
These approximate intervals are, respectively, $(10.863,12.830)$, $(4.497,7.161)$, $(2.397,3.117)$ and $(1.548,2.631)$ years. They are asymmetric because inversion is nonlinear; positive log-rate or likelihood-based intervals could also be inverted. The symmetric delta intervals and the inverted-rate intervals are two legitimate first-order approaches and need not have numerically identical endpoints.
The death state has zero exit rate, so its <holding time> is infinite. There is no finite mean-sojourn estimate or ordinary finite <confidence interval> for this structurally <absorbing state>. All intervals above concern the mean parameter, not a <prediction interval> for one patient's realized <holding time>.
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