Hong Kong is a very beautiful and unique city. Thirteen hours makes a nice amount of time to explore -- Especially if you've never been before, I think it would be a mistake to skip entirely just because protests are happening.
Allow yourself extra time to return before your flight, and go out during daylight if possible. You might avoid wearing black, which is the color protesters wear. Even the most nationalistic partisan protester won't hassle you if you aren't mainland Chinese or dressed like a police officer, and the police won't hassle you unless they mistake you for a protester.
It should be simple enough to avoid getting caught in the middle of things -- if you see a mass of people in the street, walk the other way. (Or think of it as an experience and watch from the sidelines! Limit your picture-taking, which can irk police.)
From the airport, Kowloon is closer than Hong Kong Island, and it seems protests most frequently happen on the island or the Southern part of Kowloon. In the more Central and Northern parts, you can go to a (less crowded!) Tim Ho Wan, find great char siu, walk through Ladies Market, Flower Market, or the Golden Computer Arcade, or enjoy parks like the Kowloon Walled City park.
Closer to Tsim Sha Tsui will be a lot of shopping, Night Market, or Chungking Mansions if you're a Wong Kar Wai fan. And as another user answered, you can also see Lantau peak very close to the airport to enjoy the curving mountainous, seaside landscape.