"Drunk with fatigue" is indeed a metaphor. The soldiers Owen is describing are not literally drunk, and of course it is not literally possible to drink fatigue! However, what Owen is suggesting is that the effect of their fatigue on them is similar to drunkenness. They are suffering through their march in a sort of stupor, perhaps slightly blurry and removed from reality.
A metaphor is a figurative device in which a comparison is drawn between one thing and another without the use of words such as "like" or "as," which we would expect to find in a simile. When a metaphor is used, the poet or writer will simply state that something IS a thing that it is not, in a literal sense. In this case, Owen states that the soldiers ARE drunk with fatigue, but context makes it evident that he does not mean this in a literal sense. Instead, he is inviting us to draw our own comparisons between the behavior of the soldiers and the behavior of a person who is drunk. He is using the comparison to convey how intensely tiring the life of these soldiers is. The idea of soldiers marching in this sort of state also poses the question of how safe it actually is for them to continue fighting under these conditions, with no respite or sleep.