Yes and no. Spiders can feel things akin to emotions, but don’t have brains developed enough to have full blown emotional capacity. As far as befriending one goes, you can try, but I’m afraid that friendship will be very one-sided. Maybe try approaching it slowly, talk to it, and try to find out what it is interested in. See if you two have something in common, maybe try to share a fly together. (I can already hear you gagging at the thought of eating fly guts) Hey, look, you’re the one trying to make friends with a spider, OK? Some sacrifices will have to be made for the sake of friendship here.
Spiders with BIG Brain… for a Spider
A spider's brain is small. Very small. Even when you consider a relatively larger spider like a tarantula. You’d figure that a larger spider has a larger brain, and that may be true in some cases, but they don’t necessarily act any more sophisticated or complex than the smaller spiders. In any case, any spider you study has a smaller brain. It doesn't have spare neurons just lying around wondering whether or not you two are friends. Even with their small brain, however, they can hunt, lay traps, stalk prey, and have a PhD in web design. Unlike your cocky neighbor that thinks he's an internet whiz because he was able to set his Wi-Fi network name to "God-Level Hacker."
Spiders Don’t Read Poetry and Get in Touch with Their Sensitive Side
Spiders don't have emotions in the same way we may consider emotions. They can sense degrees of danger, which is similar to fear; irritation if they are damaged, similar to pain; stress, which is similar to stress; and even possibly a small amount of pleasure during mating (but the purpose of the nerve endings used during mating is still unconfirmed).
They do this by use of their senses. There are two basic functions of these senses: obtaining physical information and chemical information. Some spiders have two eyes for higher-definition vision and six more for motion detection, while others are near-sighted grannies trying to knit a blanket that comes out as a spider web. Anyway, eyes, they have eyes, and those eyes detect light (physical information). They can also feel other physical things like pressure or temperature (touch) and use little hairs to sense sound waves (not quite hearing, but hey, close enough). They sense chemicals by use of their little hairs as well, which would be the closest they come to taste and smell.
A spider doesn’t really have a need for complex emotions. It may, over time, become accustomed to you as the caretaker and then instantly forget you. Or it could be fine with your hand meandering into its cage and dropping off crickets or other food, or cleaning the place up, and may even come to sense that your hand is safer to climb onto because you’ve never hurt it, but do know that it will always see you as a potential predator simply because of your size. Yeah, that spider doesn’t care about body shaming, and it will always see you as “the big one.”