How do I point a subdomain to a service outside of Vercel?

Following this tutorial: https://vercel.com/support/articles/pointing-subdomains-to-external-services

There are two things here: set A record and CNAME record in Vercel. Should I set both?

If so how should I replace the "example" field for CNAME?

I did for A record, but get an error:

Do I need to set middleware settings in my backend? I was not expected it, as locally I did setup a proxy and it works locally.

app.middleware.use(CORSMiddleware(configuration: .init(
    allowedOrigin: .all,//.originBased,
    allowedMethods: [.GET, .POST, .PUT, .OPTIONS, .DELETE, .PATCH],
    allowedHeaders: [.accept, .authorization, .contentType, .origin, .xRequestedWith, .userAgent, .accessControlAllowOrigin, .init("crossDomain")/*, .accessControlAllowCredentials, .xRequestedWith*/]
)))

// next.config.js

/** @type {import('next').NextConfig} */
const nextConfig = {
  reactStrictMode: true,
  async rewrites() {
    return [
      {
        source: "/api/:path*",
        destination: `${
          process.env.NEXT_PUBLIC_ENVIRONMENT == "PROD"
            ? process.env.NEXT_PUBLIC_BACKEND_URL
            : "http://localhost:8080/"
        }:path*`,
      },
    ];
  },
};

module.exports = nextConfig;

I tried replace "example" with "api", not possible:

Existing record conflicts..