If it's an exact copy, then there's nothing you need to do really.
In reality, you may have to change the ip address, but that won't affect JIRA directly. If the server name has to change though, you'll want to amend the base url in JIRA.
I would test the basic connectivity of course - can you still get to JIRA on the url you want to use. Obviously there's some generalised testing you can do (create, edit, comment etc), but with a direct copy, I doubt you'd find problems at that level. Check that the emails going out have the right urls on them, and I'd focus testing on integrations with other systems - check all your application links and addons that may be linking JIRA to other places.
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