Yes, "As soon as we arrived home,...." is grammatically correct. You may recall commonly used words like "home" often are expections to the genteral rule, so that we say "get home", "go home", and "arrived home", but must normally use a preposition with others nouns. Michael Swan, in his book PRACTICAL ENGLISH USAGE, includes "home" in his list of adverb particles (other adverb particles includes "above" ,"around","off","out" and so on).