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.
SUPPLEMENTAL to Maciej Stachowski's ANSWER
Cool as Slang
This use of cool appears to have arisen among urban African-Americans in the 1930s. It was widely adopted among admirers of cutting-edge jazz (the 'beats') shortly after WWII, and spread in the 1950s into the speech by which young people (those in their teens and early twenties) distinguish themselves from their elders—which is what we usually mean by the term 'slang'.
Cool was virtually universal in my generation, the Boomer cohort. It has remained in use among us, and has been received into the speech of succeeding generations: I hear it a score of times a day in my shop and from my Millenial son and his friends.
If you come to the U.S. you will find many, many elderly people—me, for instance—employing cool as a universal approbative.
So I think cool in this sense has long since passed out of 'slang' and become an ordinary mainstream colloquial usage. Indeed, I think we can pin down just when this happened: in the mid-80s, when the surfer speech community started preposing their fresh slang intensive way. Clearly, cool was no longer sufficiently marked to represent the language of an in-group. By 1990 the chat-room l33t found it necessary to invent a new spelling, kewl, to indicate that their use of cool was ironic.
Best Answer
Lit in the modern vernacular means not only "intoxicated" (or "lit up") but has come to be a positive modifier which occupies roughly the same semantic space occupied over the years by such terms as cool and groovy, and swingin(g).
This very day in the New York Times, the "Quotation of the Day" is:
If you use the expression as in your question, though, you will tell your listener that you (the plural you) intend to become intoxicated this weekend. If you want to communicate instead that the party will be enjoyable, you might express it as: