About 5 years ago, I had a changce to get a reservation at a hotel where I really wanted to stay.

If I sent my money on time, there werould be no problem tomaking a reserveation.

"Reserve" generally has an object after it, so you could say "to reserve a spot" or "to make a reservation."

My correction above (changing "were" to "would be") makes the sentence grammatical. However, given the context of your story, I would suggest one of the sentences below:
If I had sent my money on time, there would not have been a problem making a reservation.
If I had sent my money on time, there should not have been a problem making a reservation.

These two sentences indicate that there was a problem making the reservation before you explain what the problem was. The first half of the sentence indicates why there was a problem (money wasn't sent on time).

But it failed because mythe bank app on my cell phone made errors.

At lastIn the end, Mmy reservation was cancelled and I was depressed by it.

"At last" is often used to say that something has happened after you have waited for it or hoped for it. A more neutral phrase would be "in the end."

But they couldn’t find why it was agure out the problem.

Splitting this sentence from the last one is fine in conversational English, but otherwise starting a sentence with "but" makes it an incomplete sentence. The same is true with your other sentence starting with "but" above. I would suggest joining this sentence with the one before:

I claimed the error to the bank, but they couldn't figure out the problem.

In the end, i was disappointed with how the bank handlinged the problem.