How often do comets survive passage by the Sun?
Key factors:


How close is perihelion? Too close and it may be destroyed on its first pass. We know Halley's Comet, which has a perihelion of about 0.6AU, has been orbiting for over 2000 years, passing the sun every 74-76 years and is still going strong.
How big is it? Every pass loses material, so a bigger comet could last longer.
What is its composition? Darker comets can absorb more energy when close to the sun, which may accelerate outgassing and breakup.


And of course some comets hit large objects - Shoemaker-Levy hit Jupiter and was destroyed.