"عِنْد شادي بَيْت."
Translation:Shadi has a house.
I'm just a beginner, but A.S. Tritton (Arabic. Teach Yourself Books, p. 35), indicates three ways, one of which is with the preposition عند. Another is through the preposition من. Another is the usual way in Semitic languages, viz. by appending a suffix to a noun. Here's a link to the latter: http://www.languageguide.org/arabic/grammar/possession.jsp Apparently there are other ways, such as the famous expression الإضَافَةُ. There's a ton of information on the internet on this subject but it's too advanced for me at this point.
It's actually a subject, believe it or not! The sentence literally says "At Shadi is a house." The house is the thing that is at Shadi, so it's the grammatical subject. If it were an object, it would have still ended in -an, but the reason it's not "baytun" is that at the end of an utterance, we don't pronounce word endings (at least the ones that are indicated only by diacritics and not letters).
You are right about "Shadi" not receiving grammatical endings, but that's because it ends in a long "ii" sound, and that sound has its own rules in this regard. But other names can receive endings.