In order to be there on time, I got up earlier than usual.

I'm not sure what you mean, but maybe: I'm going to be there [ in / on ] time [ as / because ] I woke up earlier than [ usual / I usually do ].

This sentence has two independent clause with a cause/effect thought. There are two ways to link these thoughts with because/since conjunction. Because= focuses on the cause of why something happened. Since= this is more formal than because and it puts focus on the result and/or time element. Example: I am going to be there on time because I woke up earlier than usual. Because I woke up earlier than usual, I will be there on time. Since I woke up earlier, I am going to be there on time. I am going to be there on time , since I woke up earlier than usual.

"Wake" is not perhaps the best verb to use here. "Wake" has two problems. First, waking does not mean getting up; lying in bed awake doesn't help us get anywhere. Second, waking is an involuntary action; to arrive on time is a deliberate act. Getting up is a more logical means to the objective of "arriving on time". When should we use "wake"? Perhaps in this kind of situation: I was worried about missing my morning flight. I did not sleep well and woke up at 4 am, much earlier than usual.

Or possibly: Needing to be there on time, I woke up earlier than usual.