As has been said above, the best option is really just to stay in the city and catch a train (or, if you must, uber) to the airport.

Rydges is an option if you really want to stay at the airport, but will likely cost more than staying in the CBD. Aerotel is likely more expensive than Rydges (it's really intended for transit passengers, with it's hook being that it's physically inside the airport and does rooms per-hour).

Australia isn't like the US in that most hotels near the airport won't have a shuttle, and even if they do, it won't be free. The Novotel at Wolli Creek is likely the nearest hotel to the airport (that isn't at the airport), and has a paid shuttle.

However if you really want to move, I'd suggest something near Mascot train station. I've stayed at the Meriton Suites on Coward st, and friends have stayed at the Meriton Suites Mascot Central - both are good. There's also a Holiday Inn and a number of others around. And it's an easy trip from there to the International terminal on the train (or Uber). These also have the advantage of having a number of nearby restaurants. There's also a few options one stop further out at Green Square (eg, ValueSuites which is likely going to be the cheapest option you'll find near a train station)