Consider that if this is your first time in Tokyo, there's a good chance you'll get hopelessly lost initially. So either stick to a very simple plan, or make sure you have data on your smartphone, or both. You can pick up a SIM card valid for 7 days (or even less) from a vending machine or desk at the airport for a small coin. It'll take you about 10 minutes to locate and buy. Alternatively, check if your current phone provider offers data roaming in Japan for a reasonable price. Especially if you need to be back at the airport in time, you do not want to be stressed out by guessing which way you need to go, or by missing the last train. Both Google Maps and Apple Maps have excellent public transit directions in Japan.
If you're saying your plane lands at 6pm and your connection leaves at 1am, that effectively leaves about 3 hours for you to enjoy. You'll be out of the airport by ~7pm, it'll take at least 30m to go anywhere, you'll need to return the same way, and you'll have to be back by ~11pm. So that leaves 7:30pm ~ 10:30pm under ideal circumstances. Probably less.
If you're saying you need to be back at the airport by 1am, that leaves a bit more time; but consider that the last train will leave before midnight, so realistically that'd only give you about one additional hour.
So what can you do with your 3-4 hours? Certainly not more than get a small whiff of Tokyo life. Do not try to pack too much into your visit. Pick one area, wander around a bit, get something to eat, and combine that with either drinking, some quick karaoke, or just more wandering around. That's pretty much all you'll be able to do realistically.
You'll mostly be confined to the southern-ish parts of the city. Going to anywhere like Ueno or Ikebukuro, you'll just be wasting too much time on the train.
Even Shinjuku will take you almost an hour to get to, and since Shinjuku station is notoriously confusing, it'll definitely be a full hour before you're out. You can consider the Airport Limousine Bus, which can get there in 30-40 minutes. But it's not as frequent as trains, and it may get stuck in traffic, and don't forget you'll also need to get back, which is too uncertain when you need to ensure you've reserved a ticket and you need to find the bus stop in time. If loud and busy and confusing and way too big is what you're going for, Shinjuku is it; but personally I try to avoid the area as much as possible.
Shibuya is certainly a cool area to visit. It's almost as far from the airport as Shinjuku, just about 10 minutes closer. It'll take you 40-50 minutes to get there. But it's a lot more navigable than Shinjuku. You can cross the famous crossing and then immediately lose yourself among the big and small eateries and bars in the area. And after a few drinks you'll have just about enough time to head back the same way, if you can find your way back to the station.
The next stop after Shibuya would already be Harajuku, which is smaller, less busy, and more trendy. I would probably not recommend you try to walk from Shibuya to Harajuku, as it's not the most interesting walk, and you'll just waste time.
Somewhat off the beaten path would be Shimbashi. It's mostly a business area during the day, but has many small eateries. Walk out of Shimbashi station westwards towards Nishi-Shimbashi, and you'll have plenty of opportunities to eat and drink. Again, you'll mostly encounter salary men coming from work, so it's not exactly party central. But the more you get towards Nishi-Shimbashi, you'll find good beer places (e.g. Craft Beer Market) spilling out onto the street. What it definitely has going for it is that it's only 30 minutes from Haneda airport, directly on the Keikyū-Keisei Express (pay attention that you get a train with through service to the Asakusa line; the Keikyū line goes in many different directions in many different ways).
Be warned that the areas around Shimbashi like Ginza, Tokyo station or Shiodome are mostly dead at night.
You could get off one stop earlier at Daimon. It's not a particularly interesting area in itself, but you can walk down straight to Zojo-ji temple and Tokyo Tower behind it, and there are still plenty of decent eateries near the station. Consider that the Zojo-ji temple itself is closed at night; I believe the park around it would be open though. If you're going here, this would be the only time to consider the monorail and go to Hamamatsuchō, which is right next to Daimon. Other than this, the monorail doesn't go anywhere interesting and will just waste your time.
Consider going for a shabu-shabu restaurant. Nabe-zo is a very good chain (there are of course others as well). They have a restaurant in Kawasaki, just 13 minutes on the train from Haneda. It's not quite Tokyo, but won't make too much of a difference, if you're going for the fairly unique experience of a shabu-shabu instead. You'll still find enough Japanese charm and karaoke places around Kawasaki station to fill another hour if you need to. Of course, you can also go for Nabe-zo Shibuya, to combine both experiences, if you have the time. You should reserve in advance.