Save My Phone

Two days before my phone broke (또는 died) when I was emailing to someone because of (my) work.

I felt down terrible and I could do noting except (just) wait until the next morning.

The next morning, I went to the woolworths to buy a micro simcard because I've used an iPhone, and it needs a micro simcard to be activated.

But there was no a micro simcard because it was sold-out.

I felt down terrible again and I asked to an old-woman who seemed like a staff member, "Isn't there another way to get the micro-simcard?"

But there were no micro simcards there either and I gave up buying it.

Just at that moment, I saw a list of the shops that sell the micro-simcards by accident; the electronics store Dick Smith was on the list.

Saving My Phone

Gerunds are common in titles.

Two days before ago, (<- put the comma here) my phone died when I was sending an e-mail to / e-mailing someone because of my work.

"Two days ago" = Your phone died on May 7

I felt terribly down terrible and I can do nothing can't do anything except just wait until the next morning.

"terribly down"? or "down and terrible"?

"can't do anything" is more commonly used than "can do nothing". Also take note that "can't do nothing" is grammatically wrong, but you can hear it in slang.

"can't do anything but/except [verb]"
"can't help [verb-ing]"
I think you got confused with these two expressions.

The next morning, I went to the Woolworths to buy a micro simcard because I've used an iPhone, and it needs a micro simcard to activate it be activated.

The last "it" can be confusing. "It" can be "iPhone" or "micro simcard". So just change the verb form to remove the confusing "it".

I felt terribly down / down and terrible again and I asked to an old woman who seemed like one of the staff, "Isn't there another way to get the a micro-simcard?"

The simcard is not specific, so use "a". It is like saying, "Just ANY one piece of micro-simcard..."

She said, pointing toward somewhere, "You can get the a micro-simcard right over there."

There was "The Big W", toward where she pointed.

But there weren't any micro simcards / there were no micro simcards as well and so (more logical than "and")I gave up buying looking for it.

"gave up buying" means that you already have the simcard, and you were about to pay for it, then you gave up.

Just at that moment, I accidentally saw / I saw by accident a list of the shops that sell the micro-simcards by accident, and there was the electronics store, Dick Smith, on that list.

What you wrote can give the meaning of "sell (the micro-simcards) by accident".

Put the adverb near the actual verb. "I saw by accident"

At that moment, I felt I was on top of the world.