Learn Dutch: De bemanning kijkt omhoog naar de horizon.

Omhoog describes a direction of movement or a gaze directed upwards. Boven describes a static location, meaning "above" or "upstairs." Because the crew is directing their eyes upward, we use the directional adverb omhoog rather than the absolute position boven.

Doesn't naar usually mean "to"? Why is it translated as "at" here?

While naar is the standard preposition for moving "to" a destination, it is also a fixed preposition for the verb kijken (to look). In Dutch, you always say kijken naar for "looking at" something or someone.

Since a crew consists of multiple people, why is the verb singular (kijkt)?

Bemanning is a collective noun. Just like with de groep (the group) or het team (the team), Dutch treats the word itself as grammatically singular, even though it represents multiple people. Therefore, it takes the singular verb kijkt.

Could I change the word order and put omhoog at the very end of the sentence?

It is much more natural to keep omhoog next to the verb. Kijkt omhoog acts almost like a single unit here ("looks up"). In Dutch, a short directional adverb like omhoog usually comes before a prepositional phrase like naar de horizon.