Well, this depends on what you mean by "justified"; it is understandable that a person like you describe in your question would doubt that the Earth is a sphere; that doesn't make her right, though.
One is justified in being skeptic of any accounts that seem to go against common sense and common experience, but one is not justified in presuming that their particular set of experiences should prevail over the experiences of others. If someone comes to me and tells something like, "in Turkey women are forbidden from walking on the streets without male company", I am entitled to doubt this until more evidence is given; I am not entitled to an argument from ignorance, such as "I have never seen or heard of any place where such a rule holds, consequently the claim is false". I am entitled, however, to ask something like, "yes? In this case, who buys groceries for the house?" (Or, of course, to google for "Turkish women walking alone on streets", which would by the way quickly dispell the whole xenophobic tale.)
Similarly, your farmer is entitled to ask, "if the Earth is really a sphere, how does it happen that things don't slide and fall from it?". He's also entitled to dispute your evidence. Yes, boats may disappear in the horizon (and then come back, proving that they didn't simply sink), but this is compatible with many Earth configurations besides a sphere; Earth could be, say, a paraboloid. It isn't reasonable, though, to argue that people must be lying about circumnavigation or space travel - here Occam's razor would start working against the hypotethical farmer; the complexity of a theory that explains why people would lie about such an issue is probably much greater than the complexity of gravitation. The farmer could be still "justified", in the sence that his surprise is understandable, if this is the first time he hears about it; his skepticism isn't justifiable if his children's school, the TV, the books he access to, the internet, are all agreeing that the Earth is a sphere.
Plus, Occam's razor should be used when theorising about a set of given facts, which are no longer under discussion. We know that things fall to the ground, or anyway to the nearest surface below them. There is no dispute about that fact. Now philosopher X theorises that this is because angels push things down. Occam's razor should be used to shave this hypothesis out, because entia non sunt multiplicanda praeter necessitatem, and philosopher X is creating an entity (angels) that aren't necessary to explain why things fall (as everybody since Aristotle knows, this happens because things have a natural place according to their composition, and things predominantly made of earth have a lower natural place than things predominantly made of water, or air, or fire...)
But when the facts are in dispute, we would need a different razor; Occam's doesn't trim that kind of hair. If the farmer denies that boats disappear in the horizon without sinking, we will need to show him this curious fact (or he will need to show us that boats in fact do not disappear in the horizon), which is something Occam cannot help us with, at least not if only armed with his razor.
The problem, of course, is that not all people have access to all facts. A hunter gatherer living in the Magdalenian culture would not have the necessary information about ships disappearing in the horizon, much less circumnavigation or photographs of the Earth taken from space. His shaman would perhaps have noticed that the shadow in the Moon during an eclipse looks like an arc of circle (though he would probably not realise it was the Earth's shadow), but the common warrior/hunter would have probably never noticed it. So his ignorance would be historically "justified"; we do not know things that we cannot know due to the technological historical constraints we are subjected to. An Athenian philosopher, or even an Athenian sailor, would be much less justified; they already knew about disappearing ships, and the philosopher at least would certainly understand that an eclypse is caused by the Earth coming between Moon and Sun (that would be known at least since Babylonian astrologers investigated the issue); moreover, the Greek philosopher would know of a much better argument for the sphericity of Earth than disappearing ships, circular shadows, circumnavigation, or space photos, which I discuss below.
The way it was first demonstrated that Earth is a sphere was by measuring shadows. Greek philosophers measured the shadow of some standard object at about the same time (at least, the same day) in different places (Alexandria and Athens, IIRC), and those shadows have a different lenght that can be only explained if the Earth's surface is curve (they assumed, I think, that such curve was a sphere, out of superstitious beliefs about spheres being "natural" or "perfect" solids; but the experience can be repeated by measuring shadows in more than two different places, and in this case other curved surfaces would be necessarily dispelled out of simple geometrical calculations). This is a much better argument, because anyone with the necessary foreign contacts and communication tools can repeat the experiment at a desired time (while eclipses are time consuming to say the least, circumnavegation and space photos require technological advances only available much later, and disappearing ships do not demonstrate much more than a generic curvature that could even be merely a local phenomenon).