Summarize the following:
You will need a quiet, protected spot to keep the trapped cat before the clinic appointment and after the surgery. It should be a place where the temperature is mild (anesthesia renders them with less ability to regulate their body temperature), protects them from other animals. Make sure it is also  quiet and free of disturbance. A spare bedroom or any dark area of your house would be appropriate. Closets or finished basements would also do nicely. To make sure the cat will be enthusiastic for the food you put out in the trap, as well as not overfed before undergoing surgery, skip a feeding. It might be hard, but resist the urge to put food out before you're ready to trap the cat. While you can withhold food, don't withhold water! Be sure to continue to give the feral cat water even the night before the attempted trapping. At the appropriate feeding-time (ideally 12-24 hours before the appointment), fold a lightweight cloth lengthwise and insert it to cover the wire bottom of the trap and the trip plate. Put about 2 tablespoons of canned cat food (or tuna, mackerel, or some other strong-smelling enticing food) at the very back of the trap. Set the trap and put it on a flat surface where it won't shift or tip prematurely.  You can drizzle some juice or oil from the food in a zigzag pattern from the back of the trap towards the front if you want some extra attraction. Alternately, sprinkle a few grains of dry food leading into the back of the trap, but not too much. Consider putting an empty lid or cup in the cage to fill later with water after the cat's been trapped.  Make sure the container doesn't have any sharp edges that the cat could get cut on. You can use a turkey baster to fill the lit from a comfortable distance. Don't leave the trap unattended but you don't have to spring into action immediately after the cat is trapped. It can, however, calm the cat down considerably to cover the trap immediately with a blanket or tarp after you see the trap has sprung effectively.  Either way, bring the trap inside to your holding area as soon as you notice the cat's been trapped inside. It might thrash considerably, so be careful to keep your fingers out of the openings. The cat may whine and make otherwise terrible sounds that can be heartbreaking. Remember: You're doing the right thing.

Summary:
Prepare your holding area before setting the trap. Withhold food for 12 hours before the appointment. Set the trap. Wait and watch.