I think that he was thinking of Amtrak service. I guess that it depends on what kind of city you are thinking of. In other words would a large suburb count? If not then you have to look at cities like Naples, FL or Annapolis, MD. A good possiblity is a Sun Belt city grown large with retirees, or tourists but little or no industry. Naples fits this, or if CSX has pulled back from St. Petersburg. Or a gambling city Carson City, NV or Laughlin perhaps. Marin County in California perhaps.
In Canada, two provincial capitals have no rail service - Charlottetown, PEI and Saint John’s, Nfld, as both provinces are without railroads.
Regina, Sask and Calgary, Alta do not have passenger service, although Calgary has land cruise trains.
Is Calgary the only North American city that has streetcars but no passenger trains?
Albany, New York proper has no Amtrak service, you have to cross the hudson into Renselaer. As for New York City, there is indeed freight rail service into three of the five buroughs: The Bronx & Queens are served by CSX, CP, P & W and New York and Atlatic. Brooklyn is served by the New York Cross-Harbor RR and NY & A.
Naples has rail service by Seminole Gulf to the very northern edge, I believe, unless they don’t use it. On a related note, another large city up the line, Fort Myers, has no passenger service either. They do have a bus connection, though.