For instance, Boston maps to "America/New_York", Amman, Jordan maps to "Asia/Amman", etc. https://en.wikipedia.org/wiki/List_of_tz_database_time_zones#Legend
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
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.
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.