I have not done ASP.Net project publishing to production before. If you are already familiar with building and publishing ASP.Net apps using Visual Studio, then you can open the ASP.Net API project in Visual Studio to build it. You can read more on that here: Publish an ASP.NET web app - Visual Studio (Windows) | Microsoft Learn.
Once you do that and you have your API link, you can follow the process used for compiling the Quasar frontend here: Publishing your Radsystems Project - Tutorials - Radsystems Forum, by providing the URL for accessing your API and compiling it.