Are there any direct buses from Jieyang chaoshan airport to Xiamen downtown?

No, there is no direct long-distance bus that runs between these two places.
Upon landing, take shuttle bus to get to Chaoshan Railway Station first. The ticket fare is CNY 18 per person. After arriving, you can find many bullet trains that can take you to Xiamen North Railway Station. The train schedules are from 09:20 to 21:28. It takes around 1.5 hours on the way.

Thx Katherine, I arrive at Jieyang chaoshan airport at 3.10pm. I wish to take the 6pm bullet train to Xiamen. Can I make it on time ? and is it safe to buy the train ticket at the station instead of pre-booked.Tq

The Chaosan train station is near the international airport with regular coach/bus connection. As for train tickets, it depends on the date you are travelling.