In journalistic and academic writing the general rule is to write out the full name the first time a person is mentioned and you may use just the surname subsequently, assuming there is a single figure with that surname. If this person habitually uses initials rather than spelling out the full name it is acceptable to stick to initials only, for example, J.R.R. Tolkien.
In your example, I advise using "J.A. Gagarin" at first and then you may use "Gagarin" (or a pronoun).
You cannot drop the possessive simply because the noun is a proper name or a public figure/celebrity. Take a more common name for example. If you wrote, "Jeff flight was historic," that wouldn't make any sense, right? It has to be, "Jeff's flight was historic."
As for a title, you likewise couldn't say, "Dr. Smith car is parked outside." It logically must be, "Dr. Smith's car is parked outside."
If what you're asking is how to avoid using the phrase "J.A. Gagarin's flight" over and over, then there are various ways to phrase your ideas that don't become repetitive. Firstly, you don't need to clarify whose flight it was each time, assuming that there is only one pilot under discussion. Secondly, use some pronouns or synonyms rather than repeating the noun "flight" each time.