If we do one day discover intelligent alien life (or they discover us), we can probably expect some eventual cross-pollination between cultures, even though they’re likely to have very different histories before contact. Some kinds of humor are more universal than others: inside jokes may only make sense to a handful of friends, but everyone can appreciate an expertly-crafted dad joke (well, maybe not).
One way to think about this is turn it around and ask what kind of jokes would make sense if you were suddenly transported to a different planet with a different history, having only an alien-to-English phrasebook to work from. You’d need to understand them without deeply understanding the aliens’ shared cultural or historical background. This eliminates inside jokes, as well as the wider categories of satire and parody pretty quickly. Anything too topical, or based solely on prominent people or places being quirky is out.
Puns and wordplay are also on shaky ground, because they rely on specific properties of the language in which they are expressed. Even between human cultures, homophones and homonyms rarely translate well. One-liners often rely on this kind of thing, so most of those go away. It also seems exceedingly unlikely that the form of the knock-knock joke would be precisely reinvented by civilizations that grew up in parallel, without speaking.
Tonal humor such as sarcasm and deadpan is interesting. It seems unlikely to me that a randomly selected species will have anywhere close to the same signals for “I’m pretending to be serious, but I’m really not” that we have. It would very likely seem actually serious to the uninitiated (just as human jokes all too often go over human heads). Certain forms of irony may also be too subtle, but I wouldn’t be surprised if certain dramatic ironies manage to exist in wildly different cultures (e.g. the audience knows something that a character doesn’t, and they make silly mistakes because of it).
Bathroom humor and other vulgar jokes might survive if they’re sufficiently universal. It’s harder to imagine aliens being very different from us than being very similar, even though they’re likely to be quite different, so this could be a faulty assumption. (See: every movie about little green people). This might be, however, the place where you expect to see the biggest differential effect beyond what you see between two human cultures or human languages. Because we have biology as a forcing function on this kind of stuff, it’s somewhat universal species-wide.
In short, our first meetings with aliens will probably be fairly serious. (One would hope so, anyways!) We’d need to wait for cultural understanding before the really funny stuff starts happening. None of this means that aliens would be humorless: they could of course have their own puns, their own cultural references, their own wordplay, etc. that we don’t get at first either. These things would take a while for us to understand, and probably a while longer to really laugh at since understanding a joke doesn’t automatically make it funny. If we make it that far though, and human-alien friendship lasts that long, there’d probably still be plenty to chuckle at in this crazy cosmos.