He's a growing cat, he's going to be playful, feisty, and push his limits.
You have to show him dominance so that he knows who is in control while also not being too hard on him and making him fear you.
When Gigi mentioned a cat tree, she meant a
cat tree that you can buy for your cat.. which, if you can find one small enough it should fit in your apartment fine.
Cats love high places. They love being up high enough to feel large/dominant and be able to observe the things around them.
My cat loves to sit on top of my refrigerator for some reason. *shrugs* I would really prefer if he didn't go up there, but no amount of punishment has made him stay down, so I just let him sit there sometimes if he wants to (there is nothing there that he can really mess up or anything that can hurt him).
Your kitty needs some kind of toys that catch his interest though! Have you tried toys with feathers or maybe the ones that crinkle like bags? Cats sometimes like toys that dangle on strings. My cats love bags and boxes too! You just have to keep trying out different things until you find something that he likes.
Kittens always play roughly. They, usually, eventually learn to be more gentle and retract their claws when playing, but not always. The best way to help with the pain here, for now, is to keep his claws trimmed. If you don't know how to do it then a vet will usually do it for you. I think it costs about $10-15 extra to get my cats claws trimmed when he has a vet visit.
Lastly, he is a baby, you have to be patient with him. He will still be pushing his limits when he goes into his teen/young adult years.. but even humans are like that. You need to be consistent with what he can and can not do so that he learns what is bad and what isn't. You need to be patient as well as stern so that he knows when he is in trouble and he knows when something is okay for him to do.
Having a pet is a lot of work. It takes a lot of time and patience. They are very lovable when trained correctly, but they are time consuming and will try your patience at times.
My result to keep mine from destroying things while we were away or being lonely/crying when he was alone? We got another cat to keep him company. Since you can't do this, you will have to make sure he has toys and things that he enjoys playing with and you will have to spend time with him and pay attention to him. I'm glad that you decided to get a pet, I just hope that you are prepared to take on the responsibility of owning one.
He looks really big to just be around 4 months btw.. well, except the last picture.. but maybe my cat is just small or something because he was about half (maybe a little over) that size at that age. o-o;;