Losmen Fadel 

This is a perfect little place to base yourself in Yogya. It's a 5-minute walk from the train and bus stations. I was offered a pickup from the airport as well. It's also a 2-minute walk from the main street, Malioboro, so you have all the food and shopping you need just around the corner. Some historical spots are at the end of that street. The hotel is a cute little place run by a family. They'll rent you a scooter and do you laundry should you need it. Simple breakfast as well. It's located in a small alley so even though my room had a balcony there was no view (which I don't care about). The WiFi in my room was strong. Drawbacks: due to it being in a local house there was no hot water (that's life in the tropics). It's close to a mosque so prepare to hear the Call to Prayer at 4:30am. Sometimes the kids downstairs were a bit loud, but it wasn't a big deal. I'll stay here again.