Fair question. I suppose "arrive home" works because "home" in this example is not a specific place, but a relative place. Your home is not my home.  This also applies to "go to Paris" vs "go abroad". You may be right in thinking it works as an adverb.

However, "his home" is a specific place (and clearly a noun). So, we do use a preposition.

You can hear both expressions, but at least where I am from in the US, "arrive at home" is much more common, and sounds more natural in most sentences. Tge phrase alone is not enough to tell us if it sounds more natural in tgat sentence, though. We might need the full sentence. 

I think in spoken language,  "get home" may be used more often than "arrive home" or "arrive at home" either one.

So why is "at" used if either can be used? This is a case where I don't have a satisfying answer. It's just how it is normally spoken when that is used where I am from.