I'm from Philadelphia and currently live in Japan. Most years, I make a trip back to see family. So perhaps my experience will be useful to you.

Before COVID, you could sometimes get a roundtrip flight NYC-TYO for less than US$1000. Often this would be on a Chinese carrier, with a layover in Beijing or Shanghai. But since COVID, I haven't seen any of those flights available. Also, things like fuel costs and inflation generally have added to the cost of flights. So I've never seen a flight for less than that in the last few years. And actually, prices are often above US$2000 when flying with a nicer airline (JAL, ANA, etc.) or when going at a more peak time, like the holidays.

So I think getting a flight for less than US$1000 is not going to happen. But there are still some things you can do to try to minimize cost.

Others have mentioned Skyscanner and Google Flights. Those work well for comparison purposes. You can also set up alerts with your desired itinerary, dates, etc.

Note that you should always book through the airline itself. Never book through a third-party site. If you have any issues, you want to be able to contact the airline directly and have them work it out. Go check out the r/flights sub for plenty of horror stories from people who didn't follow this advice.

There are still some lower-cost carriers running routes, like Asiana. When I did a quick check with Skyscanner, they were the cheapest at US$1100 roundtrip for JFK-NRT via ICN July 7 to July 21.

You might also sometimes find cheaper fares through split-tickets. That is, flying with one carrier from NYC to some layover location, then flying with a second carrier from that layover location on to TYO. Or maybe even more than one layover. But I would advise against this. First, these are usually booked through third-party booking sites, which I've already mentioned a bad idea. But even if you book them directly, they are on separate tickets. So if something happens to delay the first flight such that you're late for the second, the second carrier is under no obligation to rebook your ticket to another flight. It's a big gamble. Also, this probably means long layovers. But then you spend a good chunk of your vacation in airports and on airplanes instead of at your destination. NYC to TYO direct is around 14 hours. I've seen itineraries that take more than 24 hours to make that trip with multiple layovers. If your time is valuable to you, you'll avoid these.

Speaking of decreasing the travel time, you should fly out of NYC (any of the three airports). There are no direct flights to Japan from PHL. And few to Asia. So you'll probably have to connect through another place in North America, like Toronto, Chicago, or Dallas. In my experience, flights in and out of these airports are much more likely to be delayed than those in and out of Asian hubs.

Lastly, you do know that the average temperature in Japan during June and July is about on par with the surface of the sun, right? Not to mention the humidity. If you have any flexibility here, I highly recommend going in the spring or fall.