I just afraid since I have to be at the Frankfurt airport by 6:30 am
that might not be a possibility
Why would that not be a posssibility? According to the Bahn website, for one weekday in Sept 23, there is an S8 S-Bahn leaving Mainz Hbf at 6:02, getting to FRA Regionalbhf at 6:28, and regional train, R3, leaving Mainz Hbf 4 minutes earlier, at 5:58, getting to FRA at 6:19.
Use the Bahn schedule website, put in Mainz Hbf to Frankfurt Flughafen Regionalbahnhof, your date, and arrival time of 6:30. Looks like there are about 4 trains per hour taking 21-26 minutes.
In the past, I've stayed at Königshof, which is a short walk, right across the Bahnhof Platz from the Hbf. They have great breakfasts, but you will probably be leaving too early for breakfast.
You probably want to stay away from the über expensive hotels in the airport itself, but there are many hotels in Kelsterbach and Gateway Gardens (look them up on Google Maps). Some are probably more reasonable, and there is an S-Bahn station in Gateway Gardens as well as short bus routes to the airport. Or, if you are light packers, it's less than a half hour walk.