Localization and globalization is covered in the official docs.
Globalization and localization in ASP.NET Core
The browser passes culture information to the a web application. Web API does not have a user interface so for a multi-culture Web API you'll need to find a way to pass the culture information to Web API. Maybe use routes to drive the culture for the current request.
Since you get to pass whatever error message back to the client, that bit of logic should not be a problem.
Use the standard model validation pattern found in every beginning level tutorial.
Model Validation in ASP.NET Web API