I've never managed sysadmins but I am one, and I've had to deal with this-is-not-a-drill situations affecting hundreds of servers losing thousands of dollars a minute many times. In my experience, nothing can replace an in-depth and intuitive (i.e., coming from real understanding and experience) knowledge of the entire flow-chart (so to speak) of what happens from browser to web server and back, and then specifically what happens in a particular web application from the time a request comes in to when a response goes out.
If you find your sysadmin can't give you the entire flow, generally, from browser to server and back, after training, I'd suggest he or she is not worth keeping in a sysadmin capacity.
If I were giving this "fire drill", I'd probably leave it free-form, give a time limit, and have the sysadmin write down his/her thought process and what he/she would check from top to bottom. You can't expect perfection there, but it would be a good start to find gaps in intuitive knowledge.
Also, don't let sysadmins put themselves in a box. To say, "That's the database; the DBA should troubleshoot that while I troubleshoot other things," for instance, lets a sysadmin get away with not intuitively knowing the flow of an application from start to finish and, thus, not understanding it completely. At the very least, a sysadmin should be able to eliminate all/most other possibilities and when his/her knowledge is expended, know exactly whom to call for help. (Knowing when and whom to call for help is an indispensable skill of its own.)