Is it possible to download and overwrite a .resx file on mobile?

My game uses SmartLocalization, which uses .resx files as resources to keep the localization data.
Would it be possible to write a script that checks online for an update to the .resx and saves it on the local device?

Yes it is possible. The following methods can help you in the LanguageManager found in SmartLocalization:

This method takes the language data sent in and tries to append it to the currently loaded language.
LanguageManager.AppendLanguageWithTextData(string languageDataInResX)

This method tries to change the whole language with completely new data from the sent in .resx file.
LanguageManager.ChangeLanguageWithTextData(string languageDataInResX, string languageCode)

Hope that helps.