Do you think that Farquhar's dream made it easier for him to face death? Why?

I think that sounds like a romantic attitude--and Ambrose Bierce was certainly not a romanticist. Far from it. He was noted for being a realist, a cynic, a pessimist. If Farquhar thought he was dying for a noble cause, it didn't prevent him from having all sorts of regrets and concerns about self-preservation. In what he thought were his last moments, he was thinking about escaping and getting back to the heaven on earth represented by his beautiful wife and his lush, sunny plantation. He must have also regretted his stupidity in allowing himself to walk right into a trap with his big can of kerosene and  pocketful of kitchen matches. 

The narrator tells us in Part I what is going through Farquhar's mind:

He closed his eyes in order to fix his last thoughts upon his wife and children.

He unclosed his eyes and saw again the water below him. "If I could free my hands," he thought, "I might throw off the noose and spring into the stream. By diving I could evade the bullets and, swimming vigorously, reach the bank, take to the woods and get away home. 

As these thoughts, which have here to be set down in words, were flashed into the doomed man's brain rather than evolved from it the captain nodded to the sergeant. The sergeant stepped aside.

Farquhar is not thinking about dying for a noble cause. He is thinking about his wife and children and wishing it were only possible for him somehow to escape from this horrible situation. The last words of Part I are intended to set the reader up to believe for most of the rest of the story that what Farquhar was wishing really happened. The rope broke and he landed in the swift waters of Owl Creek. It took the soldiers some time to get organized and start firing. He managed to free himself. The reader is completely engaged in Farquhar's hopes and problems because it is natural to identify with any character when we are held in his point of view and share his motivation, which in this case is simply self-preservation, a feeling we all have all our lives. Even though we may not sympathize with Farquhar's politics and ideals, we identify with his desire to stay alive and to get back home.

He is not thinking about dying for a noble cause but about saving his life and getting back to what he has sacrificed for an idea. Ambrose Bierce, the cynic, the realist, has us believing that Farquhar has almost made it into the outstretched of his beautiful, loving wife--and then:

As he is about to clasp her he feels a stunning blow upon the back of the neck; a blinding white light blazes all about him with a sound like the shock of a cannon--then all is darkness and silence!

Peyton Farquhar was dead; his body, with a broken neck, swung gently from side to side beneath the timbers of the Owl Creek bridge.

Ambrose