I don't know. The celebrities of the future may not be movie stars and pop bands and such, but there are bound to be colorful, exceptional individuals whose exploits, accomplishments, and/or scandals attract public interest. That's pretty much a constant throughout human history. Look at Cleopatra or Lord Byron or Mark Twain or Charles Dickens or Houdini or Nellie Bly or Buffalo Bill or whomever. Can't imagine that will have changed by Kirk's time.
Indeed, according to "Amok Time," Spock had already become something of a living legend on Vulcan, which is why T'Pring did not want to marry him. Guess she didn't want the spotlight of being the wife of a celebrity!
And I imagine Lenore Karidian generated lots of press. Think about it: glamorous actress-turned-murderer tries for comeback after being "cured" by advanced 23rd century psychiatry. The tabloid headlines practically write themselves!