Tight in American English is a colloquial term that means "polished, well put together (as an act, such as a stand-up comedy routine or an ensemble's musical performance)".
I suppose the figurative meaning derives from the idea of being "taut"; there were no sections of the performance that "sagged" in terms of audience engagement or we're "loose" or "sloppy" in terms of timing.
A band-leader could say to his band when practicing a new piece: We're almost there. We just need to tighten it up in a few places.
Jack Paar was a TV host in the late 1950s on whose show comedians could appear before a national audience, and so "to be ready for" him, means that you have attained a degree of mastery of your craft sufficient to appear before a national TV audience.
You're ready for the big time!
I think he's ready for the pros. Let's bring him up out of the farm league.