The Moxy is not far from the train, either (the train is called the "S-Bahn"). As you exit the Moxy, walk across the street into that little park. The station is almost straight ahead, it's a walk of 1 or 2 minutes.

A day pass good for unlimited rides in Frankfurt city and the airport area which includes all local trains, subways, and local busses costs EUR 9.75. A single ride between airport (or Gateway Gardens) and the city costs EUR 5.00. A single ride for the short three-minute hop between Gateway Gardens and FRA's regional train station is EUR 1.50 (select "short-distance" ticket).

While I've never stayed at the Moxy personally, I've seen their shuttle bus making the rounds. I'd probably still take the train as it runs 5-6 times per hour and is faster than the courtesy shuttle.