To be able to serve my website over HTTPS on the custom domain I need
to somehow apply my certificate. How can I do it?
Unfortunately it is not possible as of writing this answer because Azure Storage still doesn't have the ability to allow you to bring your own certificates. You will need to use CDN if you want HTTPS enabled custom domain mapped to your static website hosted in Azure Blob Storage.
From this link
:
It's easier to enable HTTP access for your custom domain, because
Azure Storage natively supports it. To enable HTTPS, you'll have to
use Azure CDN because Azure Storage does not yet natively support
HTTPS with custom domains. see Map a custom domain to an Azure Blob
Storage endpoint for step-by-step guidance.