20 hour layover in Copenhagen - what's doable

You can leave your bag in a locker at the airport.
https://www.visitcopenhagen.com/copenhagen/luggage-storage-and-transportation-services.

There is train to the center of town takes about 15 minutes.
https://www.visitcopenhagen.com/copenhagen-tourist

Just be back at the airport 2 hours before your flight the next day. That is a reasonable time window

It is a reasonable time window, as Jazz said, Since it is a domestic flight from Rome to Copenhagen, you won't have any delay once you deplane.

But what's doable depends on the time frame. What time do you arrive at CPH? And where do you plan to spend the night?

Thanks. I arrive early in the afternoon and depart the next day, so I thought I'd either return to the airport in the evening and kip in the airport or if one of two friends have space, surf their couch and head back in the morning.

That's probably enough time to see Nyhavn, the Kongens Have (gardens adjacent to Rosenborg Castle), the Round Tower, Kongens Nytorv, and Christiansborg Slot, grab a Smørrebrød for a quick meal, and maybe take a stroll on the Strøget if you move quickly. Those sights are all within a fairly compact area. You might not get to all of them depending on your pace. I wouldn't suggest using any of your limited time to see the Little Mermaid.

These are the pretty standard tourist sights in Copenhagen (there are others, like Tivoli and Christiania, but unless these specifically interest you, I think your time is too limited for them). If you prefer, you could try to get off the tourist path, and your friend can probably give you some other suggestions.

From Copenhagen Airport you have the choice of two modes of transport. Regular train and the Metro. If you want to see the main tourist sights that Lane mentions, then go for the Metro, getting off at either Kongens Nytorv or Nørreport.