>(I'm a NEET)
Now see, this is the crux of the issue. If you just hang in your apartment all by your lonesome, you'll lose the battle over and over again. I suspect you already know this, even if you've not really consciously realized it. You need some activity that requires you to leave the house and socialize. A good entry-level one would be jogging, which would also help you get rid of the weight. It might not seem to help much with the socializing aspect, but you never know. The road has always been a good place for chance encounters, no?
>Of course, I cannot go to sleep without a fap
Unless you're being literal and you can't even fall asleep after a good 24 hours of wakefulness without the ol' Rosie Palms, that's not an issue. In fact, it's one of the best possible applications of masturbation, a natural sleeping aid. I know this board has plenty of abrahamic moralists who'd prefer you abstain completely, but I consider them just as deluded as the ones who know no moderation on the other end of the spectrum.
>The thing is though, I don't consider myself addicted to masturbation.
Everyone always prattles on about how saying something like that is a clear indication of addiction. While true in some instances, it is no universal rule. I'd say it indeed doesn't apply to yours. You've simply fallen into the standard trap of having nothing better to do.
>It's not the fapping that is the problem, but easy access to porn.
If you know how to set up a filter, you also know how to knock it down, so it won't help you. Other than to remind you why you put it up in the first place, and sometimes, that's all you need. Regardless, you can't get around the fact that it's easily accessible. You'll need to willfully ignore it and decide to do something else.
If the jogging idea doesn't work for you, due to laziness or poor weather (I know it wouldn't work for me due to those exact reasons), you could also just play more vidya. I know, I know, folks think that's bad too, but it's more useful, improves hand-eye coordination and all that jazz. Try out a more wide variety of games, knock up the difficulty of the ones you already play. I'd recommend trying some of the old greats, since they're cheap, challenging and fairly easily accessible these days. I suspect you might have quite a few of them already, though.
Here's a couple of tips: Freespace 2 Source Code Project, updates an old classic, often called the pinnacle of the space dogfight sims and offers plenty of high-quality fan productions to add at your leisure.
Thief 2, great game in it's own right, has had a solid decade of a dedicated fanbase cranking out fanmissions, some of which are arguably even better than anything the base game offers.
Heroes of Might and Magic, I personally like 2 the best for nostalgic reasons, but 3 or 5 would probably be the best of the lot, objectively speaking. Easy to get into, offers a good degree of replayability.
Another thing you could try is art. Not the modern kind that's more about the bullshit explanations around the pieces than the pieces themselves, I'm talking good old fashioned, hardcore replication-of-reality kind of art. With the advent of the digital age, you don't even need to blow a huge wad of cash to get started, just get yourself a cheap drawing tablet and some basic software (Paint Tool Sai is a good choice and only costs around 50 bucks to acquire legally) and you're set. Then you just git gud at it, for extra challenge try doing nudes without getting overly aroused (it's actually quite easy, so long as you focus on mechanically replicating what you see). It gives you something to do that requires effort but isn't physically taxing, provides you a vector for socializing by way of local art clubs and if you git gud enough, you'll have a marketable skill on your résumé.
Of course, I'm an artist-in-training myself, so take that glowing recommendation with a grain of salt. Just remember that you'll hate it at first, especially if you've never been much for drawing, since you'll suck something awful at first. But I've heard it said that even the best artists have a thousand shitty pieces. So get to shitting those thousand turds, you'll crap out nothing but diamonds after that, kid!