How do I ask my boss to be in charge of scheduling when all the servers work?

Just ask. No way you get it, but you can ask.

You just go up and ask for the role (do it in private, ask for a meeting). Have a good argument ready. Odds are not good that a relatively newer person (unless everyone is 16 too) will get the job though. More responsibility is generally earned by experience and demonstration of being able to handle it.

Don't ask. Go ahead and do the schedule. Then when you are finished, go to the boss and say something like "Sir the regular scheduler left us without warning; so I took the liberty of writing up next period's schedule for your approval."

Having been a boss all my working life, I always appreciated when one of my reports came to me with the solution rather than with a problem. You go to the boss with the solution (the completed schedule) and if he's any good at being a boss, he'll love you for it.

And if he tries to get you to do it as part of your job you respond, "I'd be happy to, sir, but I'll need a pay increase to cover the extra responsibility."

You can ask, but it's highly unlikely he will choose you.

Easy, you just say " <Boss's name> will you be in charge of scheduling, now that <other person> isn't working here anymore?"

You can show some initiative. Survey your co-workers and find out what schedule works best for them. Collate all that data together to come up with your recommendation. Then provide that recommendation to your boss.

Don't expect your boss to accept your recommendations though. They're likely to think of you as being a little presumptuous.

You can ask, but you are probably the youngest and least experienced person where you work.

just ask

Tell him you are an arrogant jerk that thinks you know more than everyone else there.
At 16, they are one and the same.