"To have a crush (on someone)" is a light-hearted and slightly juvenile way to express a strong affection for someone. It is mostly used to talk about young people's feelings, and does not normally describe adult emotions.
If applied to adults it can be an insult to imply the person is or acts immature. This depends greatly on context, however, as it can also be used as a compliment to imply someone is acting or looking youthful.
A "crush" doesn't have to be on a peer or someone of similar age. A young person can have a crush on an older (even a much older) celebrity or any other person. Also, a crush can often be one-sided, implying that the affection isn't returned by the other person (referred to as "unrequited love").
By using "crush" instead of some similar term, it implies that the feeling isn't serious and is likely to be temporary. Of course many young people will disagree with this, but that's the nature of youthful crushes. No one should be ashamed of a crush, but they might be deeply embarrassed if talked about in front of them.
How young is "young" can vary widely by who is talking. At my age, I might use "crush" for anyone under 30, but I'm a grumpy old curmudgeon.
Best Answer
To catch up with X means to be up to date with X, either by getting information or performing outstanding tasks.
Caught up by itself can mean "stuck due to some circumstance outside of my control" Interestingly catch up in present/future tense doesn't ever mean this, but (will) get caught up by can be used as an equivalent. Got caught up is valid as well.
With X can be used with this sense of caught up which works well if X is a group of people and you temporarily joined or "got caught in" their group for a while unintentionally.
So this second meaning is likely intended.