I hope that that passes. = I hope that passes.
The first that is a conjunction and is optional after the verb "hope". The second that is a "demonstrative determiner". When the two "thats" are spoken together, the conjunction is reduced and very hard to hear. It is natural English intonation.
We, Brazilians, don't use passar with this meaning, unless in this form: passar com alguém.
Espero que isso não passe com ele
Espero que isso não aconteça com ele
I hope that doesn't happen to him
This seems to be a Spanish meaning. I don't know about Portuguese from Portugal
Edit: there's also the idiom passar por essa/aquela/esta situação often used when something embarassing or bad has happend to you or someone else.
Não acredito que ela me fez passar por isso (por essa situação)
I can't believe she made go through this again
The phrasal verb "end up" isn't used that way. "To end up" describes a situation or activity that ends in an unexpected and, often, unpleasant way.
Joe is going to end up in prison if he's not careful.
We had planned to drive to the beach this weekend, but we ended up staying at home because of the bad weather.
I don't think so, unless it was a very specific context where the meaning can be similar. At first thought, we normally use them like this:
Espero que isso passe - I hope this will finish/come to an end soon, I hope it won't be/take long
Espero que isso aconteca - I hope this will happen, come to pass
Esperar = to hope & to expect