The second set of instructions you have is what you'd want to use.
If you're in Route 53 you would use an A record that is an Alias to target the ALB in your environment. Make sure the security groups on your ALB are setup correctly as that may be what's preventing you from connecting to your application once you have it directed there.
If you're using something like Wordpress on the site behind the ALB you may have to turn on SSL support to get it work correctly and be all encrypted. But otherwise you should be good to go for the most part. If it's straight Apache there is a potential you may have to implement a re-write rule for SSL.
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