Cheapest transport Kagoshima to Osaka

I'm looking for the best way to travel from Kagoshima to Osaka on the 2nd/3rd of January.

Our JR pass will have expired by then, but happy to travel by train if it's viable. The train fare seems okay, but would we need to book seats for this period because it's Japanese holidays?

I know flights are sometimes cheap (couldn't find anything at this time), but it's reasonably costly to get to the airport right?

I've also read there are buses, but I couldn't find anything.

Advice would be appreciated!

The train fare seems okay, but would we need to book seats for this period because it's Japanese holidays?

It is not mandatory, but highly recommended. It may be difficult to make reservations especially in the afternoon of the 2nd and 3rd.

Usually the cheapest way from Kagoshima to Osaka is by Peach Aviation, but during the New Year holidays even their rates are not that great (beware of hidden fees):
http://book.flypeach.com/

Skymark could also be an option on the Kagoshima-Kobe route, but reservations are not possible until 2 month before travel date:
http://www.skymark.jp/en/

I know flights are sometimes cheap (couldn't find anything at this time), but it's reasonably costly to get to the airport right?

About 1000 yen.

I've also read there are buses, but I couldn't find anything.

Yes, there are overnight buses, but the competition for their tickets will also be very high around January 3:
http://www.kintetsu-bus.co.jp/highway/routelist/route_detail.php?no=30

General information on travel between Kagoshima and Osaka:
http://www.japan-guide.com/e/e4604.html?aFROM=2157_4600