Instructions
Travel to Union Cave from the side facing Azalea. Go to the left side inside the cave to find a lone rock and press "A" here. This will give you a hidden Big Pearl that is useful in battle.
Surf across a lake of water in the southern part of the cave. This leads underground to the second level of the cave.
Battle a trainer and his Pokemon in the next area. After defeating him, surf across another body of water to a second trainer. Battle him and then take the nearby staircase down to the third level of the cave.
Surf down to battle a trainer blocking your path to Lapras. If you have the inclination, you can also take on two other trainers on this level and earn some useful items for your trouble. Once you beat the trainer in the southern section of this level, surf to the right and you will see Lapras swimming in the water.
Capture Lapras by hammering it with Dark, Heavy or Ultra Balls. You can also use the False Swipe move if you have a Pokemon character with this skill.