Cars and traffic lights: I doubt this would be an issue. Sure, he doesn't know how an internal combustion engine works. Neither do 90% of the people in New York. But he's presumably quite familiar with horse-drawn wagons, and he knows not to jump in front of those. If he sees a large object moving at high speed, he doesn't have to understand how it works to know not to jump in front of it. I'd think he'd figure out traffic lights pretty quickly. He sees all these vehicles racing along and then they all stop. There are these big lights hanging above the street. How much genius does it take to notice, Hey, when the red light comes on, they all stop. Even if not, he waits until they stop before crossing the road. Even if he times it wrong and he's still in the middle of the street when the light changes, plenty of modern Americans cross streets when the light is against them. He might annoy drivers but he'd be in only slightly more danger than the average pedestrian.
Communication: Where in Europe is he from? If he speaks English, sure, the English he speaks will be very outdated, but people will understand him. He's from the age of Shakespeare and the King James Bible. Modern Americans struggle with Shakespeare and Jimmy, but they can understand them. If he speaks Spanish, there are lots of people in New York who speak Spanish. I don't know how much Spanish has changed since 1500, but if it hasn't changed more than English has, same situation. Failing that, a big city like New York is used to having tourists who don't speak the language. No one would be shocked by his speech, and people are used to trying to communicate with foreigners by pointing at things, pantomiming actions, etc. If he doesn't find someone who speaks his language he's going to have a hard time engaging in a serious philosophical discussion or getting any sort of abstract information. But he should have no problem communicating simple things like "I need food": I think pointing into your open mouth and rubbing your stomach would work in many cultures. If he runs into the authorities, they'll search for an interpreter, and finding someone who speaks German, French, or Greek wouldn't be all that hard.
Police: Assuming he's just trying to survive and get back to the exit point, I don't see why the police would bother him. He has no reason to murder or assault anybody. He's probably not selling cocaine or engaging in insider trading or operating a coal mine in violation of environmental regulations. He might look and act weird, but in a big city, there are lots of weird people wandering around. I don't see that he'd have an issue with "avoiding capture". New York police are not normally on the lookout for time travelers from the 15th century, and while New York has a lot of bizarre laws, I don't think they have one against time travel. Oh, if he's carrying a sword or some other medieval weapon, that could get him into trouble. Besides that, the only likely source of trouble I see would be if he steals food to live or breaks into a building to get shelter from the weather.
Which brings us to, food and shelter: As others have pointed out, water wouldn't be much of a problem: there are plenty of decorative fountains and public water fountains in New York. It wouldn't take long to figure them out. Food is harder. He could try begging. There are plenty of homeless beggars in New York: no one would find this startling. Someone mentioned that he would surely recognize cathedrals, maybe other churches if they have a prominent cross. Someone at a church would surely try to help him. Churches in big American cities routinely expect to help poor and homeless people.
Beyond that, we're back to communication. If he is able to communicate, if he speaks English or Spanish or manages to find people who speak his language, he might learn that some random object he is carrying is a valuable antique that can be pawned. Any coins he is carrying would surely have value. If he asked around he might find an odd job to earn some money. He might run into difficulty that he can't legally work in the U.S. because he doesn't have a social security card, but plenty of businesses hire people under the table to avoid taxes and regulations. If he can't communicate, getting any sort of work would be hard.
With no identification, he might get into trouble as an illegal alien. But American police don't lock someone up just for being an illegal alien. He'd be told he has to appear in court on such-and-such a date, and he'd be gone long before then.
He might have the strange problem that if people tried to help I'm by giving him money, he wouldn't know what it was. Paper money was not introduced in Europe until the 17th century. If he can communicate, no problem: He says, "Huh, what's this?" and they say, "That's money. Don't you recognize American money?" He might struggle with the concept but, etc. But if he can't communicate, it's just an odd-looking piece of paper. Why are people he asks for help giving him pieces of paper?
If he tells people that he's a time traveler from the past, they might think he's nuts, but other than telling each other, "Hey, this guy's nuts", I don't see any more coming of it. It's illegal in the United States to lock someone in a mental hospital unless you can prove that he is a "danger to himself or others". If he's not threatening to kill himself or attacking other people, no one's going to lock him up. There are plenty of people in New York who think they've been kidnapped by aliens or are the rightful heir of some fortune. No one worries about them much.
It suddenly occurs to me that you didn't specify whether he knew he was going to travel to the future and had time to prepare, or if this happened to him unexpectedly. If he had time to prepare he could bring along a week's worth of food, make sure he had clothing suitable to the season, have some gold or silver for trade, maybe even bring a tent. Then he just camps out in Central Park or wherever for the week and it's no trouble at all.