What does "hard pushed in argument" mean in this context?

Question

In "In the Midst of Alarms" (1894) by Robert Barr, a blacksmith, Macdonald, did an embarrassing trick to break the conceit of his New Yorker client in front of the crowd in his smithy, but he failed

Macdonald saw there was no triumph over him among his crowd, for they all evidently felt as much involved in the failure of Sandy’s trick as he did himself; but he was sure that in future some man, hard pushed in argument, would fling the New Yorker at him. In the crisis he showed the instinct of a Napoleon.

“Well, boys,” he cried, “fun’s fun, but I’ve got to work. I have to earn my living, anyhow.”

Yates enjoyed his victory; they wouldn’t try “getting at” him again, he said to himself.

Macdonald strode to the forge and took out the bar of white-hot iron. He gave a scarcely perceptible nod to Sandy, who, ever ready with tobacco juice, spat with great directness on the top of the anvil. Macdonald placed the hot iron on the spot, and quickly smote it a stalwart blow with the heavy hammer. The result was appalling. An instantaneous spreading fan of apparently molten iron lit up the place as if it were a flash of lightning. There was a crash like the bursting of a cannon. The shop was filled for a moment with a shower of brilliant sparks, that flew like meteors to every corner of the place. Everyone was prepared for the explosion except Yates. He sprang back with a cry, tripped, and, without having time to get the use of his hands to ease his fall, tumbled and rolled to the horses’ heels. The animals, frightened by the report, stamped around; and Yates had to hustle on his hands and knees to safer quarters, exhibiting more celerity than dignity. The blacksmith never smiled, but everyone else roared. The reputation of the country was safe. Sandy doubled himself up in his boisterous mirth.

“There’s no one like the old man!” he shouted. “Oh, lordy! lordy! He’s all wool, and a yard wide.”

Yates picked himself up and dusted himself off, laughing with the rest of them.

“If I ever knew that trick before, I had forgotten it. That’s one on me, as this youth in spasms said a moment ago. Blacksmith, shake! I’ll treat the crowd, if there’s a place handy.”

1- I found that "hard pushed" means "face a great difficulty in doing something". However, I'm having difficulty understanding the full meaning of the phrase in this context

2- What did he exactly mean by "I’ll treat the crowd, if there’s a place handy"?

Answer 1

Your understanding of "hard pushed" is correct. "hard pushed in argument" means "facing great difficulty in putting his side of the argument".

"fling the New Yorker at him" is a bit more difficult to explain. Imagine that the Blacksmith is having a heated argument with another individual. It's common in this situation to bring up embarrassing or difficult incidents for the other party in order to put them on the back foot. What the Blacksmith fears is that, in the future, someone will remind him of this incident ("fling" it at him) to gain the upper hand in a dispute.

"I'll treat the crowd, if there's a place handy" is an invitation. Pleased with the outcome, Yates is offering to buy everyone in the crowd a treat - most likely an alcoholic drink. That's implied by "this one's on me," a phrase often used when someone is offering to buy a round of drinks at a bar. "If there's a place handy" is a rider: he can only buy everyone a drink if there's a bar (a "place") nearby (i.e it is "handy").