I would really say try not to worry too much about it. Flying with a 4 month old should be a lot easier than flying with an older mobile baby or toddler. I would very strongly recommend a wrap (not a carrier, it will likely be too bulky for you to sit down in a narrow plane seat in). Wear baby as much as possible, which will be comforting, and when you can and if you need to, get up and move around with him in the wrap. More than likely, he'll just go to sleep! Planes are very soothing to babies because of the white noise and vibrations. As long as you stay calm, he'll be chill too. They don't really understand what it is at that age and as long as they're close to you, they'll be happy. Even my 2 year old, who totally understood what was going on, didn't flinch at any of the bumps or weird noises.
I would bring whatever you usually would bring on a long car journey - nappies, formula and bottles if you bottle feed, a few changes of clothes, maybe one for you too in case there are any accidents, something warm for him in case it's cold on there, a comforting toy if he has something special, etc. You can bring a pushchair to the gate with you and then they'll check it underneath the plane, but honestly I would probably just use a wrap the whole time and check the pushchair at check-in if you don't need it. It was more of a hassle dragging it around everywhere than just carrying my daughter. Pack what you need, but pack lightly. If your partner can carry your carry-on stuff, great, if you're flying alone, I found using a backpack for everything was easier so I had my hands totally free.
I've only flown with a toddler, but in comparison, a baby would be a dream! It's different when they don't understand they need to sit still and want to run around even though you're about to take off. In that case, yes, bring loads of things for entertainment. We brought sticker books and a few favourite stories to read and a few special toys. But you don't need that stuff for a 4 month old. Just cuddles. Honestly though, it was a lot easier and a lot less stressful than I thought it would be. My daughter enjoyed the plane and I wouldn't be as stressed out to do it again next time.
Oh, and get an aisle seat. Last thing you want is to be climbing over someone covered in baby sick to get to the toilet if you need to.
