Here, to start, is today's scripture:
"For Moses truly said unto the fathers, A prophet shall the Lord your God raise up unto you of your brethren, like unto me; him shall ye hear in all things whatsoever he shall say unto you.
And it shall come to pass, that every soul, which will not hear that prophet, shall be destroyed from among the people" Acts 3: 22-23
My first reaction was to think that he should have fasted less and read more: JS-H 1: 40 specifically identifies the prophet in question as Christ.
This missionary's midrash, then, that Joseph Smith was that prophet, was almost certainly incorrect in terms of its identification of the prophet in question. So why had he felt like the the answer to his question was yes?
One possibility, of course, was that the missionary had not actually fasted and prayed about that passage, but rather about Joseph Smith, and subsequently assigned his testimony to the wrong passage. In that case, I should be careful not to condemn him, that I be not condemned (Luke 6: 37). Maybe God gives us knowledge in a connect-the-dots way: here's a little, there's a little (2 Ne 28: 30), now draw the connecting conclusions that give it life-guiding meaning on your own. And maybe we, like kindergartners, draw our lines a little squiggly so that they go through places where they don't technically belong. Maybe the missionary didn't know that part of the line, he knew some dots and got the line wrong. And does that matter? To a great extent, yes: mistaking Joseph Smith for Christ in one place is probably going to cause you problems in others. But with sufficient humility and charity, you should be able to work through the problems you cause yourself by thinking you know things you actually don't, and it'll turn out OK in the eternal scheme of things. You'll be better off for having drawn your knowledge sloppily, as it were, than if you'd stuck to a few random dots of purer revelation and drawn no conclusions at all.
Another, more intriguing possibility was that the missionary had prayed about that passage, but God had discerned an intent behind the question to know whether Joseph Smith was a rasul (like the prophet Moses promised). Perhaps God, in order to assure the missionary that Joseph Smith was indeed the prophet of the Restoration, answered the question in ways we interpret as meaning yes. That scenario gives rise to another model for our spiritual knowledge in which God's revelations are often contextual and informed by personal intent, less manifestations of absolute truths than reassurances that He is with us and approves of the course we are about to take. That would explain how God could work in images of heaven and hell when they answered specific questions about the nature of divine justice, then show a more detailed and surprising vision of three degrees of glory to Joseph and Sidney, reserving a more complete truth beyond the scope of men's preparedness for future revelation.
This model may seem to suggest that actual our "knowing" is not knowing, that we cannot be certain of any truth spiritually after all. Maybe that's correct in the strict sense, but I think the more important lesson from a view of revelation is contextual is the need for continuing revelation to "triangulate" the truth. Maybe you asked something at one point in your life, and got an answer based on an intent you later forgot. If you rely on your mind's understanding of the soul's answer, you might run into trouble. If you keep the connection between God and your soul open, however, you can continue to ask your questions as context and intent change over the years, gradually refining your sense of truth until, in some future state, you come to a promised purity and fullness of knowledge.