Usage of 就 after a noun and before a verb
This sentence is expressing in a colloquial way, and can be interpreted as:


  你一大早（起床）就出门赶飞机，现在很累吧。


So 起床（getting up）is omitted，就 means the later thing happens right after the former one.

Or:


  你（在）一大早就出门赶飞机，现在很累吧。


在 is omitted，就 at here means the mentioned thing happened at an earlier time（eg. 我们很早以前就认识）