There are loads of places to store your luggage at Kansai airport (KIX). That said, it is not a very competitive market so do not expect prices to vary much.
One thing to be aware of is that rates are often charged in calendar day not 24-hour increments, so for example the 12 hours from 6 p.m. to 6 a.m. would count as two days. You can check this before using your chosen storage method.
There are manned left luggage booths at the far ends of both the first (ground) and fourth floor in the international terminal building. They are open daily from 6.30 a.m. to 10.30 p.m.
At one end is the airport baggage service.
JAL offers similar services at the other side. The pricing is identical.
There are lockers in a variety of locations across different floors – they are easy to find. They are also run by JAL but at 300/400/600 yen for small/medium/large lockers may be cheaper than the manned service (although the dimensions for these categories may not match those of the manned booths). They are accessible 24 hours. You can pay in cash and there is often a change machine nearby if you need to exchange a note for coins.
If you only want to store an item for less than three hours, you may want to find one of the orange flagged short term lockers, which are cheaper for short times at 100/200/300 yen per three hours for small/medium/large lockers.