There is no logical reason to capitalize horizon here.
Pay attention to where commas have been added and where they’ve been deleted.
When one city comes close to another is a long introductory adverbial phrase; although a short one (three words or fewer) may not need a comma to set if off from the rest of the sentence, longer ones do.
I changed the second sentence so it isn’t misunderstood. The original version could be taken to mean that the short period of time is used to trade both culture and trade resources — trade is an adverb in this interpretation, defining what sort of resources they’re sharing. I don’t think that’s what the writer meant, though. (Were this for an actual editing job — such as the one I’ll start work on as soon as I post this — I’d ask the author which interpretation they mean.) Deleting the comma and adding and to makes it clear that share and trade are both verbs in that sentence; culture is shared, and resources and such are traded.
The cities of the world are moving constructs. When one city comes close to another, there is a short period of time to share culture and to trade resources and such. Your city hasn’t seen another city in ten years. However, one is on the horizon. As it grows closer, you realize all its inhabitants are dead.
(I think I read this story; it sounds like part of the SF novel The Strength of Stones by Greg Bear.)