Both destinations are served by Colombo (CMB). Since there's no flight route between them, the best way to travel is by road.
Kochi
Fly to Colombo (CMB), then travel by road to Kochi. Taxis, buses, and private cabs are available from the airport.
Aruppukottai
Fly to Colombo (CMB), then travel by road to Aruppukottai. Local transport options include taxis, auto-rickshaws, and state buses.