This is an old thread but a lot of the advice is dated so I'd like to contribute for future prospectives.
Healthline is good public transportation (compared to the rest of what CLE offers), but I would advise not living in Cleveland Heights or the University Circle area where you'd have to commute all the way downtown every day without a car. It is FAR, maybe 130 blocks, and the infamous east-west divide in Cleveland makes it difficult to get from point A to point B with ease. Overlook is really more of a CWRU area... and really no Case students that I've ever known have lived there. They tend to live in Little Italy up the road, which is more affordable and less family-residential. Keep in mind that while Uni Circle is generally an oasis of affluence in an otherwise seedy area, its light rail station is one of the more dangerous ones. Since it gets dark around 5:30pm in the winters, it may be unwise for a solo student to rely on it daily.
Downtown itself has *really* exploded since this thread was first started and the occupancy rate is at something like 97%, so chances are you won't be able to find off-campus housing within walking distance of CSU without advance planning, but there are neighborhoods close by that are trendy, interesting, and very affordable (Tremont, Ohio City, and the Detroit Shoreway). Try one of those.