I'm pretty sure of this information having been through something similar a good few years back. I'd strongly advise speaking to a debt charity though, try stepchange as I'm sure they could give definitive advice.
The main problem that you have here is that you do seemingly owe the debt. Rightly or wrongly that means you can't appeal on the grounds that it isn't valid.
Your options are therefore limited. If the CCJ occurred in the last 30 days, pay it immediately to have it struck off. That's the only way to get it removed.
I assume that's not the case so you'd then need to either pay it, have it set to settled and perhaps put a note (NOC) on your credit file around what happened or just ignore it. In truth paying it will do little to help your credit rating but would stop the debt being chased and any manual underwriting when you're looking for credit in future may take it into account.