Hi Antti,
There are two ways to get this. One is to get the contentType object from the web or the list, then call get_nameResource().getValueForUICulture(lang). This get_nameResource method is not well documented. The value of "lang" is a string, for example "en-US" or "sv-SE".
The other way is to put a fake "Accept-Language" http header in your API call, essentially fooling the API into thinking that your browser is configured for that language. This only works if the user doesn't have a preferred language in their profile, which overrides the browser language setting.