Well, it’s actually not an API, but simply signed ZIP files with an optional complimentary web service, which you will need to set-up.
Documentation is here: https://developer.apple.com/library/content/documentation/UserExperience/Reference/PassKit_Bundle/Chapters/Introduction.html
The certificates required for signing can be created at no additional charge inside your Apple Developer account.
Alternatively, there are a couple of Open Source implementations of PassKIT for various platforms; simply perform a web search on ".net passkit".
If you do not want to take the trouble to run the infrastructure for yourself, there are also several commercial API providers, handling the details for you – provided you supply them a respective signing certificate for your needs. Please consult you Apple Developer Agreement for the details on this topic.
Though, if you dare to set up your own .net implementation on a server, it may possibly be worth it to consider adding native support for Microsoft’s Wallet format, which uses a pretty similar data structure – with angled instead of curly brackets… ;-)