Teachertravel8-I don't know what price you would be comfortable with but be aware apartments tend to be smaller especially the kitchens.Most of the time they don't have ovens or big refrigerators.Just make sure you check if it's got A/C, an interior or exterior one and on what floor-most don't have elevators and going up/down stairs w/groceries loses its charm very quickly!
I always stay either in the Barrio de Letras or Opera.