They are very much interchangeable here. The emphasis is on "korán" ("early"), so the exact verb is not that important. "Kelni" is the action of getting up. When the Sun rises, it takes several minutes. "Felkelni" is the total, completed action, at the end of which you are up and running.
In the case of getting up early, it does not really matter, you have to start way too early in either case. And that is the focus here.

There's a difference between waking up and getting up. Usually a few minutes.

"Kelni" (rising or waking) is a general term referring to "getting up" in the morning in this context and it's shorter so Hungarians would use that one more. "Felkelni" would still be correct, but people would wonder why you're getting so specific! :) "Felkelni" is more specific to the act of waking up from sleep. Hope this helps.