Affordable accomodation is not really a well known concept in Luxembourg unfortunately. A lot depends on your mode of transport and how long you intend to stay. If you have your own car, you have way more choices than if you rely on public transport.

A lot of people who work in Luxembourg actually live in France, Belgium or Germany and cross the border everyday, because of the difference in price for accomodation. Working near the airport, it could be a good option to find something in Germany, near Trier: the motorway from Trier to the city of Luxembourg goes past the airport.

You can always try to find something in one of the villages on the way from the airport to Betzdorf. I doubt whether you’ll have much luck, though. The best option is probably to try out the different agencies…

I would contact the tourism bureau and see if there are rooms for rent around the city.
Good luck! I envy you!