There are no hostels in City of Luxembourg, but there are other types of accommodations...

"Auberge Gaglioti
Entirely renovated, the Gaglioti inn, contiguous to Trattoria Dei Quattro, proposes its rooms to you any comfort, equipped each one with bath, television set satellite 16/9 (international programs), telephone with hot line, mini bar, dries hair and even taken for data-processing connection. And of course the possibility of benefitting from the benefits of Trattoria Dei Quattro"
At 50 m from the train station, our hotel is really near the city center (10mn walking...