he show her his one of a kind watch
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He shows it to her for two reasons; to give her a hint on the games. (The arena is in the shape of a clock) and to tell her that he is secretly working for District Thirteen to overthrow the Capitol.
Katniss is telling the story in first person
She didn't care about him, she was just surprised about what happened and how she killed him.
Katniss was sitting in a tree above the Careers. she heard a "psst" and looked back. Rue was there pointing to a trackerjacker nest. She told She told Katniss to cut it down so it would fall on the Careers