Hi!
I'm glad you're thinking about choosing Madrid for your year abroad, because it's such an amazing city. I'm Spanish myself, not from Madrid, but I've lived there for a year and totally loved it, since the first week I felt like at home.
Sohoscribbler has already given you some advice about the zones where to stay, but I cant give you some more ideas so you can choose. For sure Goya is really nice, actually I think it's one of the nicer places around Madrid but sometimes it can be a bit expensive (850 euros, you've been so lucky!). I was living for a while in Argüelles, it's really nice too, full of students, near the city centre (a 20 minutes walk aprox from Plaza de España) and affordable, you have to be lucky but you can find really good flats. Also, I lived on Tetuán/Cuatro caminos which can be fine or not at all, there are some good places and others than seem not so nice, but anything close to Bravo Murillo will do. Well, in fact I think that if you avoid the south (sorry for that, there are some nice places but still...) you will be ok!
If you have any question just ask or pm' me! I'm sure you will do great in Madrid and please don't forget to visit Templo de Debod by sunset because it's amazing and totally my favourite place in all the city.