In the battle arena, you can win much needed prizes for the game. For 32 000 BP, you can get the OMNISLASH. For 64 000 BP, you can get the W-SUMMON materia. To get into the SPECIAL TOURNAMENT you need to have won both of these.
If you have them both and the Ultima Weapon, talk to the receptionist and select special battle. After you win the special battle, you'll recieve the Final Cut Materia.
When fighting in the battle arena, you should always aim for the 'All materia is broken' slot. The more handicap you have, the harder and the more BP you get. Don't try and get those stupid HP/MP or broken weapon ones, they don't help. It's the materia slots that give you the BP.