Pre-condition your pet prior to the airline travel by letting them spend time in their travel crate/carrier.
Keep your pet calm before, during and after the flight.
Ensure your pet is healthy and is seen by his/her vet prior to the travel. This is required by the airline.

If a pet is on medications for a condition, be sure the pet is well stabilized before air travel.
An elderly pet will do fine as long as they are healthy and have very comfortable bedding in their crate/carrier.
Water is essential before, during, and after the flight for hydration.