Raleigh Apartments (that's the name, "Raleigh Apartments") at 740 Smallwood Drive have 1 bedrooms starting at $555. (Not sure if they are available.)
They are very old apartments, solidly built. Walking distance to Cameron Village, where you can shop (Harris Teeter for groceries). My son lives there in a 2-bedroom and loves it. He lived for a year or two in a newer apartment in North Raleigh near Crabtree Mall, but he then went back to Raleigh Apartments. You can have a dog up to 80 pounds in weight, too!
I think it would be very difficult to find a house to rent for $600/month. I could be wrong.