He developed his talents with the Second Sons of Essos and trained with a new weapon and method. Arya has been trained in the Bravosi sword fighting method. He is fast and agile, which will be important in fighting the mountain. Arya has enough of that, but she cannot, under any circumstances, lose her focus. She must stay focused, or she will die.
Can he defeat the mountain? Yes, if it comes to it. But there's no point in hoping for something like that to happen. Arya needs to fight defensively at first, use his wits around him, and only then go on the attack. Only then can he have a chance of winning.
Because he's dead. No one can win if they're already dead. That's why people don't think much of his chances.
Talents and powers During her first thirty years, Arya was heavily trained in magic and swordplay, until she was powerful enough to defend Saphira's egg. She was a sword expert who, as an elf, possessed strength and speed unrivaled by any human, save Galbatorix, and subsequently, Eragon, Murtagh, and Barst. As an archer, she was capable of shooting arrows with such force that they pierced through two dragons at once.
After the fall of Gondor, when she was about sixteen years old, Arya became a member of the new order of knights known as the Lord's Guard. Here she fought bravely against the dragon Smaug until he was slain by Eglon, king of Gondor. After this event, she took up residence in the hall of her lord's family, where she remained for several months. It is during this time that she met several other elves from different branches of the elven house, including some who were quite young. Together they discussed what role, if any, she might play in the future government of the elves.
Arya was given her own room with a balcony, where she spent most days reading or gazing out into the forest. She did not join in the activities of the household, except on those occasions when it was necessary to fight off raiders or trespassers. Once every year or so, she would go back home to visit her parents' graves, but since their death she had no other reason to leave the castle.
Arya is still a northern daughter, but she has completed her training and possesses all of the talents of the faceless man. She uses these skills to play games with people to achieve her goals.
It is possible that she may have met some other students along the way or there may be others who have yet to show their abilities. But since we last saw her, she has joined the ranks of the assassins' guild known as the black jackals. This may mean that she has become one herself.
She is not alone in this world of spies and assassins, however. There are many other people with different motives who will try to take out their targets too. In fact, it is this very fact that makes the world such a dangerous place.
It is also worth mentioning that despite the fact that she is now an assassin, she has not given up on her true love Brienne of Tarth. It's possible that she may even prove useful someday...if only to get closer to him!
In the final episode of season three "The wolf and the lion", it is revealed that Arya has taken refuge with several other girls from her family's house outside of Braavos.
The Waif is envious of Arya since she sees Arya fast achieving the same level of training that she is, yet her own progress is effectively halted. The Waif also seems aware that Arya possesses a gift for finding things, suggesting she may be looking out for her. When asked if she was hungry, The Waif replies that she is always hungry.
Arya names The Waif after one of King Robert's knights who died at the Battle of Winterfell. The knight's name was Haywood and he had been traveling with Lord Eddard Stark when they were captured by slave traders near Pyke. Arya notes that no one knew what happened to him after that, but it wasn't until years later that she learned he had been thrown into a pit with any other captives who weren't worth keeping. Only three people survived this ordeal: Arya, Sansa, and The Waif.
In terms of personality, The Waif is cunning and manipulative, often trying to get others to do her dirty work while she stays safe at home. She is also clearly aware of her beauty and uses it to gain attention from men who will protect her or provide her with food. In the end, however, none of these men succeed in getting close to The Waif because she is not interested in anyone except Arya.