Is there anything that you can see inside the Port of Singapore, or a way to learn more about the port's operations?
No, you can't. It's a very, very busy working port, full of reversing trucks and cranes flinging about multi-ton containers, many of which contain valuable and/or hazardous cargo. Short of getting a job, the only way to enter is to apply for a Temporary Pass, which requires a good reason and a company already operating in the port to sponsor you.
But if you want to get close, you can go for lunch or a drink at the Republic of Singapore Yacht Club, which is (partly) open to non-members and is located right next to the Pasir Panjang bit of the port. Alternatively, take a bumboat cruise from Marina South, although I'm not entirely sure these are still operating now that they're building the new cruise centre there. Most tall buildings nearby will also have good views of the Keppel Terminal, although there's no public observation deck or anything; nobody will stop you from going up into the HDB (public housing) buildings at Spottiswoode Park though.
The best way overall, though, is to watch this episode of National Geographic's Megastructures on YouTube.