Yes with a Basic Cloud you get one custom domain and no backups, you would just need to update your DNS records to include our CNAMEs for this custom domain. You would submit your custom domain via the Hosting tab of the account.fusionauth.io under the Action Drop down. Then you will be shown the CNAME record for that domain and you will have to update your DNS records to include this CNAME. You can see an example of this at the doc below.
https://fusionauth.io/docs/get-started/run-in-the-cloud/cloud#custom-domains
You would not be able to use your own SSL certificates. We would handle all those in FusionAuth Cloud. We create the certificates on our end and you just need to create DNS records to validate the domain with the CNAME records.