On December 2nd, We left Canberra at 11am.

And then we were headinged to Sydney from Canberra by highway bus.

"were heading" is an imperfective and might make it sound like the verb is not gonna be completed

I didn't sleep the previous night, because I'm supposed was about to see my friend who lives in Sydney.

Present tense is better when telling stories

OWhen the bus ,started, I fell asleep when the bus has started.

No need to use bus twice also "fell asleep" is past and "has started" is present

When we arrived Sydney, I finally I met her in person.

On December 2nd, We left Canberra at 11am.

And then we were headinged to Sydney from Canberra by highway bus.

I didn't sleep the previous night, because I'm was supposed to see my friend who lives in Sydney.

So my brain was wired.

I think it's correct, but we don't really use this expression, at least not where I come from. You can also say, for example "I was excited" or "I had a lot on my mind" which sound a bit more natural.

On the bus ,I fell asleep when the bus has started moving.

On the bus can be omitted because it's clear that that's where you fell asleep.

When we arrived in Sydney, I finally I met her in person.

I saw her for the first time in five years.

Alternatively, you can combine this with the previous sentence, turning it into, "When we arrived in Sydney, I finally met her in person for the first time in five years."

On December 2nd, Wwe left Canberra at 11am.

Aand then we were headinged to Sydney from Canberra by highway bus.

I think it's better to connect this sentence to the previous one.

On the bus , I fell asleep when ithe bus has started.

When we arrived in Sydney, I finally I met her in person.

I travelled to Australia last year.

This is just a comment in case you don't know, but "travelled" is preferred by UK English and "traveled" is preferred by American English. Both are correct, just up to your own style of English.

On December 2nd, Wwe left Canberra 11 am.

I didn't sleep the previous night, because I'm was supposed to see my friend who lives in Sydney.

So my brain was wired.

I would prefer "So I was wired" here.

On the bus ,I fell asleep wheon the bus haswhen it started.

When we arrived in Sydney, I finally I met her in person.

I saw her for the first time in five years.

Your previous sentence makes it sound like you are meeting her in real life for the first time ever. The second sentence, however, makes it sound like you saw her in person five years ago and are just now seeing her again. If you never met her before this, I would phrase it as "I saw her for the first time after knowing her for five years."

On December 2nd, Wwe left Canberra at 11am.

And then we were headinged to Sydney from Canberra by highway buscoach.

I didn't sleep the previous night, because I'm was supposed to see my friend who lives in Sydney.

On the bus ,I fell asleep wheon the bus has soon as it started.

(I changed this sentence, so you only use 'bus', once)

When we arrived Sydney, I finally I met her in person.