Kansai Airport to Takayama

Can anyone advise if this correct and economical train to take from Kansai Airport to Takayama
1) from Kansai Airport to Shin-Osaka by Ltd Exp Haruka
2) from Shin Osaka to Nagoya by Shinkansen Nozomi
3) from Nagoya to Takayama Station by Ltd Exp Hida

This is the best way to travel between Kansai Airport and Takayama, but it is also an expensive one. If a low cost is your highest preference, then you should consider taking a local train or bus into central Osaka and then the expressway bus to Takayama:
https://www.japan-guide.com/e/e5908.html?aFROM=2157_5900

There is very special ways to go as below:

Take Kanku Express departing at 6:37 a.m. from JR Kansaikuko Station, which arrives at Osaka Station at 7:41 a.m.
Then take JR Limited Express "Wide View Hida" departing at 7:58 a.m. from Osaka Station, which arrives at 12:23 p.m. in Takayama.

Cheapest way is to take the Nankai express to Namba and transfer to the Express Nohi Bus from Namba to Takayama. That's going to be about 6 hours and 5600Y. There are also only 4 buses per day, so depending on your schedule you may wait hours for the next bus.

Fastest way is to take the Haruka to Shin-Osaka, board the Shinkansen to Nagoya, and then transfer to the Hida train to Takayama. That's about 13000Y and 4.5 hours. Many trains per day, so you should have minimal waiting.

If you have JR pass, the train is the obvious choice. And if you plan to go anywhere else beyond Takayama, it will probably pay for itself.