Super close to the night market and right in front of a bus station, Anximen Hostel has a great location. It has lovely new facilities, including a roof garden and a (loud) bar on the same floor as the dorms. Rooms look smart and are cleaned daily. Private rooms are particularly great value; dorms are basic but secure.
Staff don't speak much English but communicate via apps and can arrange group trips to the grottoes and Shaolin Temple.
If you fancy street food for breakfast, a handful of places pop up each morning on Jifu Jie (集市街), 50m west of the hostel.