No it wouldn't because once you've matched speed with it, you stop accelerating and therefore need no more fuel. The comet isn't doing anything except following gravity.

The comet probably doesn't have a great amount of fuel. Even if it's all ice, how are you going to split the water? Only if you have a fusion reactor that can do H-H fusion, or you can scrape enough dueterium out of it, can you use that. You will find no tritium as it has a short half life, and He-3 (probably) won't have been implanted by solar wind in interstellar space, so if you need that, you have to bring it or breed it somehow.

It does have a great amount of mass, so you could rendezvous, construct a mass driver or ion drive and start taking it apart and using chunks of it as extra reaction mass. That would allow you to essentially get a free reaction mass "refill". You will still need a lot (a lot) of power, and solar will be near zero.