The “PUMA Match Summary” files available in the CITY comparability section include both the IPUMS CITY codes and the Census Bureau’s Place FIPS codes. Those files enable you to link the census city-level summary data to IPUMS microdata. Please see @Ivan_Strahof’s answer to your other post for more details on this.
As @Chris_B mentioned, the issue is that cities are not directly identified in the microdata after 1980. You can read more about this in the Comparability Section for CITY.