How large is this rock we are talking about? If it's not too big, I wouldn't worry too much about the aptasia. You can kill them off with kalkwasser paste, vinegar, or a host of other products. You are always going to run into a pest here and there. It's inevitable, if you are adding things to the tank.
Meeanwhile, cycle the tank as afore mentioned. Don't add any livestock during this process. it's perfectly normal to see ammonia for a week or more. The length of time you see the ammonia directly correlates with the amount of dead stuff on the rock you bought.
Since no two tanks, rocks, ammonia spikes, amounts of die off, or water parameters are alike, I disagree with adding a piece of cycled rock to the tank and resting assured that your cycle is over.