We booked the Hotel Burg through one of the discount websites hotelclub.net see here and got a much reduced rate from the hotels own webpage: Burg Hotel. The prices we saw were advertised for 89 euros, and we got a great room for 50 euros (that's about $60).
Burg Hotel is situated in the castle district of Buda and a perfect location for seeing the history, as it's right within the castle walls. Our room had a splendid view of St Matthias Church, which it's advertised that all the rooms do. This was gorgeous by day or night when it was lit up. Across the street is the fisherman's bastion, so you really can't get a better location for sightseeing.
The only downside with the location were the church bells at 6am. They would ring over and over again!
Our room was big and clean. Unfortunately, there was construction going on whilst we were there, which made for a very noisy room. Luckily, we were never in the room to really get bothered by this, but believe me, it was loud. Maybe that's why the rooms were cheaper.
A bus runs into Pest (to the metro stop) every 15 minutes, so getting around wasn't a problem, but the majority of public transport ended at midnight, so if you are night owls, then this could be a problem unless you are willing to risk the taxis (they can be expensive).
Breakfast was included in our price, and the breakfast room had the same stunning view out over the church. The food was plentiful, with the more traditional cold cuts of meat, cheese, yoghurts, and cereals, as well as all the hot "English-style" foods such as saugages, eggs, and bacon. DELICIOUS!