Yes, both of your sentences are correct. "Stagger" can be used both in a literal sense (having trouble walking) and in a metaphorical sense (struggling to grow a new business).
Both of your sentences are clear in their meaning, but I think in order to be 100% correct, you need make some tweaks. "Stagger" means most commonly (in my personal experience) "to walk or cause to walk unsteadily as if about to fall". It has a lot of nuances in its meaning, but for the most part, someone who is staggering is already standing up and walking (though not well).
For that reason, I would be more likely to say:
After breaking his leg, he struggled to his feet and staggered home.
Staggering is more what he would do after he managed to get himself standing.
After going broke, he struggled to get back on his feet. He started a new business that staggered forward.
n.b.: Be careful in the second sentence: "staggering" can be an adjective, meaning "causing great astonishment, amazement, or dismay; overwhelming". Starting a "staggering new business" probably means that the business is unexpectedly doing very well, not that it is struggling. You want to be sure to include a word like "forward" that shows movement, like the business is going forward on shaky legs.
Best Answer
Option 1 is correct for your desired meaning: Mr. A will be responsible for paying all the costs associated with making the new Bill book. So if Mr. B orders a new book, and Mr. C drives 10 miles to pick it up, Mr. A will pay for the book and the travel expenses.
Option 2 is also a valid sentence, but it does not have the same meaning; in this sentence, Mr. A will personally perform all the spending. Mr B. will not order the new book, Mr. A has to do that. Mr. C will not go pick it up, Mr. A has to do that. However, since there is no mention of who is ultimately responsible for the actual payment, when all is said and done, Mr. A might still be able to submit an expense report and get the company to pay for everything.