What can I do at Hong Kong Airport for 13 hours?

There are more options of restaurants etc. (at generally better prices) if you pass through security and immigration. Only passengers can enter the terminal at the moment (documents and tickets are checked outside the terminal building) so there is very little chance of disruption within the terminal buildings.

You’re a few days too early to see Lion King at Asiaworld (which is a convention center that is basically at the airport - one short stop (only HK$6) on the Airport Express away from Central- and actually walkable if you're so inclined).

For what it's worth, the risks of going into the city are well within my personal tolerance level (I was there twice last month and did not directly encounter any issues except some delays on the MTR), but I would leave more time getting back than usual.

There is also the Disney park which is near the airport (taxi is probably easiest).

All the tourism businesses in Hong Kong are hurting badly at the moment, I'm sure your presence would be appreciated greatly.

While I understand your reluctance to go to downtown Hong Kong at this time, one of the major tourist attractions is right by the airport. You can get there without entering or passing through any of the main commercial, educational or similar areas that have been the focus of protest, or taking public transit.

The attraction is the Ngong Ping cable car that takes you to the village and temple of Ngong Ping and the Big Buddha statue. You can get there by blue taxi in a few minutes. It's often visited by people on layover or about to depart from the airport.

Consider going to Macau:

It seems that the risk the OP is mentioning for not leaving Hong Kong Airport is the 2019 Hong Kong protests. While that risk is in reality pretty close to none, if you still do not wish to enter Hong Kong, then you can instead go to Macau. Macau is quite small so within ~10 hours you'll have easily seen most major landmarks.

One may get from Hong Kong to Macau (and vice versa) with the 24-hour bridge shuttle bus via the HZMB (Hong Kong–Zhuhai–Macau Bridge), which takes about 30 minutes. For more information, see Hong Kong Airport – Macau – Hong Kong. Door-to-door from the airport's exit door to Macau's territory should take between 60 to 90 minutes. Note that if you take the bus, you have to pass the Hong Kong immigration. If that's an issue, you can take the ferry to Macau straight from the airport instead.

You could sleep in a "relaxation chamber"

... which is not the same as checking into a hotel; nor is it one of the airport lounges.

Quoting from the Sleeping in Airports guide for Hong Kong:

Refreshhh by Aerotel – Location: Terminal 1, Airside, near Gates 35 and 60. Private “relaxation cabins” for single passengers that can rented in blocks of 1 – 6 hours. cabins come equipped with showers. Accessible to Terminal 1 passengers in transit only. Bookings by email: hello.hkg@myaerotel.com .

I've not actually tried that. Also, I'm not sure what the prices are, but hopefully it should be much cheaper than a proper hotel room.

Note: Check the availability and register for a spot beforehand. @jcaron suggests their HKG location may be closed for refurbishment at the time of writing.

Definitely worthwhile to look into the airport lounges. Depending on your budget, some offer very comfortable seating, where you can even take a nap, a hot shower, free snacks and beverages, and even a meal. The lounge I used had facilities equivalent to a luxury hotel.

Certain credit cards give you benefits and discounts in some of the lounges, but they should accept all other cards as well. You might be referring to the Lounge Buddy app that recently began to work exclusively with American Express. From my experience, ordering directly through the lounge’s website there’s no such restriction, although it’s more expensive.

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.