Is it necessary? No, but it is a good idea. You may have to go to several locations, lugging your luggage to each one until you find one in your budget. "Hostales" are the cheapest private lodging option in Madrid. Doubles can start at 35 Euros with shared bathroom.
It's always a good idea to visit the websites of a number of them so you can know their location, prices, and see photos of their rooms. That way, when you find one you recognize, if they have a room available, you'll know what you're getting into before entering or going upstairs to the room.
Best of luck and enjoy your time in Madrid!
Saludos, MadridMan
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