I'm not sure how long you plan to stay-n-study in Madrid starting in June but for long-term stays (like months) I wouldn't attempt to find your apartment-to-be on the internet. If you only plan to spend a few weeks fine, but for multi-month-stays I'd recommend waiting until arriving in Madrid to get something. You can't
see roaches on websites (roaches are not common in Madrid except in the oldest, least maintained buildinges - I haven't seen a cockroach yet in my 14 months of living here) nor can you tell if the people advertising them are honest - most are, of course, but you'll know better once you meet them face-to-face.
Have you tried
http://www.Habitamos.com yet? There, mostly, individual apartments are posted by owners but they never have websites nor photos. This is a good source for when you're here, though.
Your language school SURE has contacts and recommendations for lodgings for their students, don't they? THAT, neither, guarantees nice "digs", however, but your chances are probably better if you're hoping to get something SOLID before arriving - but it'll still be "sight unseen".
Check out
http://www.VenAMiCasa.com for "Home Stays" throughout Madrid and Spain. I've YET to read ANY reviews of this website or the bedrooms people rent in people's homes but it seems to be an interesting and affordable concept. Many/Most of the rooms cost as little as 12 Euros per night up to about 24 Euros per night. (600 Euros, your monthly budget, would put you at 20 Euros per night)
Anyone else have ideas? Best of luck, cja3882! Se you in Madrid!!
Saludos, MadridMan