We (https://opencagedata.com/demo) return that as part of our geocoding results. If you have only a couple of hundred cities it's still within the free trial plan

        "timezone": {
          "name": "Asia/Amman",
          "now_in_dst": 0,
          "offset_sec": 7200,
          "offset_string": 200,
          "short_name": "EET"
        },

As detaro wrote the timezone-boundary-builder project is the best data source for the shapefiles of timezones.

For a list of places the files on http://download.geonames.org/export/dump/ have a column for the timezone name. Creative Commons Attribution 4.0 License.

There are places, usually unhabitated who don't have a good timezone name, but just 'GMT-10' or such. With cities you likely don't encounter this. Short names, e.g. EET, aren't unique. There are several who share the same abbreviation.