Will a 1 amp solar charger effectively charge a marine battery and how long will it take?

If it produces 1 amp, it does not matter whether it is solar or not. How big is the battery? It should have the max amp.hours marked on the label. This will give you an idea of how many hours at 1 amp would be required. Also depends on the state of charge when you start. Chargers powered from the mains would probably give 4 to 6 amps, so it will be a lot slower.