I would recommend not going the 15 km east from FRA airport to downtown Frankfurt, but instead taking the commuter train (S-Bahn) (which stops two escalator rides directly below the arrivals area) 25 km west to the city of Mainz. The S8 line (not the S9 line, the other line stopping in the station) will take you to either Mainz Hbf (the main train station in Mainz) or Mainz-Römisches Theater, which is closer to the Hyatt. The Hyatt is just two blocks east of the Römisches Theater train station (just under half an hour trip from the airport, trains run every half-hour), on the banks of the Rhine--you can get a room with a Rhine view for 160 EUR. The hotel restaurant is one of the better restaurants in Mainz, too. At the Römisches Theater train station, you can see the ruins of a Roman theater (how the station got its name), and on one of the two blocks between the station and the hotel there is a museum displaying the remains of Roman ships and other memorabilia concerning boat travel in Roman times (free admission, closed Mondays). Mainz also has a 1000-year-old cathedral, the Gutenberg museum, and lots of wine bars (center of German wine producing area).

Another option to stay in Mainz is the Hilton on the banks of the Rhine--right now their main lobby is being renovated; the entrance is on the back side of the annex building across the street, but there is a pedestrian walkway connecting the two buildings. This is somewhat cheaper than the Hyatt (older building, that's why the lobby is being renovated) (non-refundable special rate right now of 87.50 EUR). To reach the hotel, you can either take the S8 to Mainz Hbf, or the S9, which stops just across the river from Mainz at the "Mainz-Kastel" train station. From either train station, you can take Bus 54, 55, 56, or 57 to the stop "Brückenplatz" which is right at the hotel. If you don't have much luggage, you can also just walk across the bridge from Mainz-Kastel--the Hilton is at the base of the bridge on your left as you cross the river, but the bridge is probably a ten-minute walk or so. The hotel restaurant here is also ok.

Or you could stay at the airport Sheraton and just make a day/evening expedition to Mainz. Other places to eat might include http://www.ambassenheimerhof.de/ (just off of Schillerplatz square, not far from St. Stephan's church with the Chagall stained glass windows, this is where Laura Bush ate when her husband and the German chancellor had a summit meeting in Mainz), or, for a great view across the Rhine towards Mainz, there is http://www.bastion-von-schoenborn.de/ -- this is located adjacent to the Mainz-Kastel train station.