U.S passport holder flying from Seoul to Singapore, cruise ending in Beijing

So you will arrive by cruise from Hong Kong to Beijing, and then leave for Seoul by air on the same day? I think you are eligible for the 24-hour free transit. If you want to visit the Great Wall, you need to apply for a temporary stay permit at the immigration check upon arrival. You can double check this with the cruise company. It's possible that the cruise company may insist you applying for a visa.