Answer
$t = 1.2\times 10^{-6}~s$
Work Step by Step
Let $u'$ be the speed of the micrometeorite relative to the reference frame
Then $u' = 0.82c$
Let $v$ be the speed of the reference frame relative to the ship
Then $v = 0.82c$
We can find $u$:
$u = \frac{u'+v}{1+u'~v/c^2}$
$u = \frac{0.82c+0.82c}{1+(0.82c)(0.82c)/c^2}$
$u = 0.98c$
The speed of the micrometeorite relative to the ship is $0.98~c$
We can find the time it takes to pass the ship:
$t = \frac{350~m}{(0.98)(3.0\times 10^8~m/s)}$
$t = 1.2\times 10^{-6}~s$
