Answer
${\bf 16.67}\;\rm \mu s$
Work Step by Step
Let's assume that the Earth's frame is $\rm S$ and the rocket Sirius’s frame is $\rm S'$, where $\rm S'$ moves at a speed of $v=0.6c$, and Orion rocket moves at a speed of $u=0.8c$. Both $v$ and $u$ are relative to $\rm S$.
Now we need to use the Lorentz velocity transformation equation to find $u'$ which is the speed of Orion in $\rm S'$ frame.
$$u'=\dfrac{u-v}{1-\dfrac{uv}{c^2}}$$
Plug the known;
$$u'=\dfrac{0.8c-0.6c}{1-\dfrac{(0.8)(0.6)c^2}{c^2}}=\frac{5c}{13}\;\rm m/s$$
Now we need to find the length of the Orion rocket as measured in $\rm S′$;
$$L'=L\sqrt{1-\dfrac{u'^2}{c^2}}$$
Plug the known;
$$L'=1000\sqrt{1-\dfrac{\left[\frac{5}{13}\right]^2c^2}{c^2}}=\bf 923.1\;\rm m$$
So for Orion to take completely past Sirius, it had to travel a distance of 1000+923.1=1923.1 m, at a speed of $u'$.
Thus,
$$u'=\dfrac{L_{tot}}{\Delta t}\Rightarrow \Delta t=\dfrac{L_{tot}}{ u'}$$
$$\Delta t=\dfrac{1000+923.1}{\frac{5}{13}(3\times 10^8)}$$
$$\Delta t=\color{red}{\bf 16.67}\;\rm \mu s$$
