Learn Spanish (Latin America): El chico lee la invitación solo en su cuarto. - The boy reads the invitation alone in his room.

Yes. Because solo (alone) describes el chico (the boy), it is masculine and ends in -o. If a girl were reading alone, the word would change to match her: La chica lee la invitación sola.

If he weren't alone, how would I change this sentence to say he reads the invitation "with me"?

You would remove solo and use the special pronoun conmigo (with me). The sentence would become: El chico lee la invitación conmigo. If he were reading it with you, you would use contigo.

Why use "su" for "his"? Doesn't "su" also mean "your" or "her"?

Yes, su is a multi-purpose word! It can mean his, her, its, your (formal), or their. Spanish relies heavily on context to figure out who the owner is. Since the subject of our sentence is el chico, it is natural to understand that su cuarto means "his room."

I learned that "cuatro" means four. Is "cuarto" a typo here?

It is not a typo, but they are related! Cuatro means "four," while cuarto means "fourth" or a "quarter." Calling a bedroom a cuarto comes from the idea of a room being a "quarter" or section of a house. It is used just as commonly as habitación to mean "room" or "bedroom."