You can create a JQL query that looks something like this:
Project = <my project> ORDER BY created DESC
This will give you a list of all issues in the project, having the last created issue on top. In issue navigator, display the created column in your search results and you have your answer.
If you want to take it a step further, you can even save your filter and use it as an issue source for a Filter Results gadget on a Dashboard.
Oh, I see. Your problem actually is not so much about how to retrieve the most recent issue and its create date, but about the permissions.
In order to find issues in a certain project, you need to have access to it. More specifically the 'Browse Project' permission.
As JIRA Administrator, you are usually the one who is consulted when there is a problem with the configuration of a project or your JIRA instance at larger scale. You should also be in charge of the governance of your instance. So there should be nothing wrong with having access to the projects, without you requiring to actually participate.
Although it is best practice to configure your permission schemes using Project Roles, we often make one exception by granting the jira-administrators group Administer Project / Browse Project permission through the permission schemes of our projects.
For more information on permissions, check out https://confluence.atlassian.com/jira062/managing-project-permissions-588582096.html.
As soon as you have granted yourself the right permissions, the JQL should return results.
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