I have been one acquainted with the night (1 and 14)
Though "acquainted" involves knowing someone or something, it doesn't mean being friends, or being intimate with something. Also, the speaker has been "one" acquainted with the night – one is a lonely word in both sound and meaning. So, from the start, the poem has a distant and eerie tone. On top of it, the night is a pretty lonely thing to be acquainted with. It might be less lonely to not be acquainted with the night.
I have outwalked the furthest city light (3)
So now that we've seen the emotional isolation of being acquainted with the night, we see that the speaker is physically isolated. He's outwalked the furthest city light – so he's all alone, beyond the guidance of civilization.
I have passed by the watchman on his beat And dropped my eyes, unwilling to explain (5-6)
Don't get too excited about seeing another human being, because it only makes the speaker more isolated. Policemen tend to scare us even when we're doing nothing wrong, and that's what the watchman does to our speaker in these lines. But when the speaker drops his eyes and walks on by, we're a little disappointed that he remains alone.
When far away an interrupted cry Came over houses from another street,
But not to call me back or say good-by (8-10)
We have another a chance to break the isolation – but, again, it fails. This voice is isolated and interrupted itself, traveling over houses to reach our speaker. Yet it's not calling for him, revealing our false hope that maybe someone cared about this poor guy.
And further still at an unearthly height, One luminary clock against the sky (11-12)
We already know that the narrator has walked past the city lights, but now we see something that is as far away and isolated as possible – the moon. These lines take the poem from a lonely and isolated night on earth to the loneliness and isolation of the sky. The moon is alone in the sky without even any stars to keep it company, at least not as we know from this poem.