Can I make a Freedom of Information request to find out what someone was arrested for 15 years ago?

In the UK, FOI does not give you access to person information about other people. You can only get information about the activities of public organisations - person information is confidential, and is protected by the Data Protection laws, so the authorities would be acting illegally if they gave you information about someone else's convictions.

No not under FOI.

You would have to find out from court records and 15 years is rather a long time.

Much of that stuff is available on line. Many courts have records available where you can look up a person and see the cases. Newspapers often have police dockets listing arrests each day. State department of corrections often list current and former inmates. You need to do as much research on the internet first.

Are you in the UK or US?

If you're in the US it's public information. You can get all that info from the state website. You don't need a Freedom of Information request, you just fill out the form.

There are plenty of background check websites, which might be slightly easier to use but they are going to charge you money for the same thing you can get for free.

If he was charged there should be a court record that is a public document.

It would be cool if you could!