How long will a 12v 1A battery last on a 12v 550mA fish finder?

There's no such thing as a "1A" battery. If you look at it more closely,

you may find that it's rated "1AH". In that case, the math says that

after being fully charged, that battery would run that fish-finder for

1hour 49minutes, although the battery voltage might begin to sag

toward the end of that period.

If the battery doesn't say "1AH" on it, then ignore this whole discussion.

There's not enough information to answer the question.