I need to book one for a couple nights, once when one of our group leaves and again when we leave the country a week later.
Inexpensive and airport seldom go together. The airport is very near to the city and is only 11 min. away by subway. Unless there is a very good reason for being close to the airport - like your flight leaving at 0600, come into town.
Use www.hrs.com to find a reasonably priced hotel. I usually recommend Hotel Monopol, Excelsior, Concorde, or Ibis on the Friedensbruche since their rooms will run about 65-75 € for a double.
Please let me know if you have any more questions about staying in Frankfurt. If you do not want to stay near the train station, there are some lovely neighborhoods to stay in too.
When you say Gasthaus, I think of an intimate, family run establishment, very traditional. I don't think you will find anything like that near the airport. My favorite hotel near the airport, Königshof, in Mainz (www.hotel-koenigshof-mainz.de), is more of a Hotel than a Gasthaus, but it is fairly inexpensive and right across the Bahnhofsplatz from the station and the S-Bahn into the airport.
If you have a little more time, I'd stay down on the middle Rhein, St. Goar or Bacharach. You can get in from Bacharach in just over an hour. There is, for instance, a train that leaves Bacharach at 7:30, changes in Bingen, and arrives at FRA at 8:34. There are a lot of nice, traditional Gasthäuser type places in Bacharach.
We stayed at the Albatross Hotel near the Frankfurt Airport four years ago and we're staying there again next month. They will pick us up and return us to the airport and they include breakfast. There are several walking paths nearby and four years ago we were able to walk to a lake for a swim.