Hitching a lift on a comet is an interesting idea. It gives you a lot of fuel. However there are some pretty big downsides.
First Delta-V: Getting to a comet is hard, but doable, Rosetta proves we can do it. Getting out of the solar system is harder but also do-able see the Voyager probes, the trick here will be finding a comet with an orbit that's easy to modify to take you where you want to go. You need to find a reasonably eccentric comet with the argument of periapsis pointing at the star you want to head towards. You can then use the comet as fuel to make a big burn when near periapsis to push the apoapsis out until you break orbit around the sun and "whizz" off to your destination.
Second fuel: While you will have a reasonably large amount of it the rocket equation is a problem. Because you have so much fuel you need a lot of it to move it. 234
The other problem is the fuel you have is going to be very low quality. It will be all mixed up, hydrocarbons, water, carbon dioxide, hydrogen, and dust. It will need processing very carefully before feeding into your engines. You will probably have more than one engine, each designed to burn/expel a different type of fuel. Making the big burn at periapsis is useful here because you can make use of solar energy as you make this burn.
A better option may be to link up with a comet in your generation ship process the comet into fuel. Then when you reach the right point of the orbit detach and make the burn. This way you are only trying to accelerate the mass of the ship not the ship and a dirty great comet.
Linking up to another comet when you arrive would done as a "if there happens to be a comet heading in the right direction" but should not be required as it would be impossible to know if such a comet existed in the target system with out already being there, getting the two orbits to sync up correctly would be extremely unlikely. 199