It wasn´t definitely not a good day for everyoneanyone / any of us.

The adverb goes after verb 'to be'. The word order for adverbs is difficult, because it is not the same for all adverbs. You might want to revise this grammar point. 'any' is used after a negative

We all were all supposed to check in before 6, but my brother fellhad fallen back asleep and he didn't hurry to get up.

the adverb goes after the verb 'to be' The other changes are to make the description clearer

As a result, we didn’not only leaveft our house late, but also the bus didn't passarrive/ come on time, so we arrived at the airport at 6:20, just before the check-in closed.

not only ... but also is a pattern, so don't change it

Fortunately, we couldwere able to check in and relax for a little bit, so we decided to eat something because we haven’dn't eaten anything that morning.

for the past tense of 'can', you use 'be able to'. In this sentence it sounds better with 'that morning' to qualify when you hadn't eaten

EAt that moment, everything seemed to be okay at that moment, but when we finally realiz, but by the time we noticed that theit was boarding time had come, we couldn’t help but, we had to run as fast as possible so that we cwouldn’'t losemiss the flight.

'At that moment' at the beginning is better for emphasis. finally sounds like you had waited a long time. 'couldn't help but ' usually means doing something you don't want or mean to do

Unfortunately, it was too late and we lostmissed the flight.

It wasn´t definitely wasn't a good day for everyone.

Fortunately, we could check in and relax a little bit so we decided to eat something because we haven’dn't eaten anything.

Everything seemed to be okay at that moment, but when we finally realized that the boarding time had come, we couldn’t help buthad to run as fast as possible so that we cwouldn’'t losemiss the flight.

Unfortunately, it was too late and we lostmissed the flight.