Answer
$t = 1.25~years$
Work Step by Step
Let $u'$ be the velocity of the armada relative to the reference frame S
Then $u' = 0.800c$
Let $v$ be the velocity of the reference frame S relative to the messenger
Then $v = -0.950c$
We can find $u$:
$u = \frac{u'+v}{1+u'~v/c^2}$
$u = \frac{0.800c-0.950c}{1+(0.800c)(-0.950c)/c^2}$
$u = -0.625c$
The speed of the messenger relative to the armada is $0.625~c$
We can measure the length of the armada according to the messenger:
$L = L_0~\sqrt{1-\beta^2}$
$L = (1.00~ly)~\sqrt{1-(0.625~c/c)^2}$
$L = 0.781~ly$
We can find the time it takes the messenger to pass the armada:
$t = \frac{0.781~ly}{0.625c}$
$t = 1.25~years$
