Start here. This is the direct spoken answer to practice first.
Why this question matters
A production domain changes how TLS, proxies, cookies, CORS, callbacks, webhooks, redirects, and generated links behave. The public edge must be configured and tested as part of the application boundary rather than treated as a cosmetic DNS change.
I would treat the custom domain as part of production behavior, not as cosmetic DNS. I would verify the App Service binding, DNS records, TLS certificate, HTTPS-only behavior, redirect rules, allowed hosts, exact CORS origins, auth callback URLs, webhook URLs, and any generated absolute links. I would also smoke test the public domain from outside Azure, because the default azurewebsites.net host can hide problems that only appear on the real domain.