Oh, they would run out fuel eventually. It would just take a while. Remeber, they could draw in hydrogen from the surrounding area, but not anti-hydrogen. However, to get around that, the logical response is a large fuel tank of anti-hydrogen combined with low fuel-draw.
Again, logically, they have to run out, but it probably takes a *long* time and the fighters most likely get anti-hydrogen refuelings on a regular basis.
As for capships using the same system, it strikes me as likely that they're handled like modern nuclear carriers. They have a 'deep' refueling during refits, but for the most part, they don't need 'regular' refuelings of anti-matter.