Answer
$t = 1.60~years$
Work Step by Step
Let $u'$ be the velocity of the messenger relative to the reference frame S
Then $u' = 0.950c$
Let $v$ be the velocity of the reference frame S relative to the armada
Then $v = -0.800c$
We can find $u$:
$u = \frac{u'+v}{1+u'~v/c^2}$
$u = \frac{0.950c-0.800c}{1+(0.950c)(-0.800c)/c^2}$
$u = 0.625c$
The speed of the messenger relative to the armada is $0.625~c$
We can find the time it takes the messenger to pass the armada:
$t = \frac{1.00~ly}{0.625c}$
$t = 1.60~years$
