You should clarify what do you mean with alias. For the purpose of this answer, I'm assuming you configured
EXAMPLE.COM to map
PUPENO.com at DNS level. In other words,
EXAMPLE.COM points to the same IP of
PUPENO.COM, using an
A record or a
CNAME DNS record.
In this case, the first step should be to decrease the DNS TTL, for instance to 1 hour, and wait at least 24 hours for the change to propagate. If the original TTL was equal or greater to 86400 (24 hours), better is to wait at least 36 hours.
Then, update the DNS record of
EXAMPLE.COM to point to your new IP address (or domain) and wait for the chances to propagate.
In this case, the users will not experience any downtime because they continue to see the old site as long as their provider doesn't update the DNS cache.
If your site redirects to
PUPENO.COM, simply change the redirect target to the new URL. Again, you will not experience any downtime.