Phoenix does demonstrate tremendous heroism without being aware of it. She doesn't think of herself as being brave. She is just doing what she must do to take care of her little grandson, and she has done it many times. Her path to town is "worn," suggesting that she has traveled it many times.
Phoenix faces great danger making her trip to town. She is very old and frail and almost blind. The weather is cold because it is December, hard winter. Phoenix climbs a hill, makes her way through briars, climbs through a fence, and crosses a stream by walking on a log fallen over the water. If she had fallen into the stream, she would have frozen to death. When she falls in a ditch after the hunter's dog comes after her, the hunter has to help her up. She cannot get to her feet by herself and would have died there without his help.
When Phoenix gets to town, she is treated very hatefully by one of the women in the doctor's office. Phoenix swallows her pride and ignores the woman's insults so that she can get what she came for--her grandson's medicine. That took courage, also. Phoenix is very heroic; she risks her life and sacrifices her pride for one she loves.