Since this is so far into the future we can't really say with certainty what will be the end. Not a physicist but from what I can tell, heat death sounds is the one they lean to a bit more with the caveat we really don't know. Some recent big bounce theoretical papers came out and I get the feeling that physicists leaned against this for some reasons. One of the issues is we can only go by the observations we can take right now. If the contraction started a million years ago we would not be able to observe it yet as no signals would reach us for a long long time as it takes so long for light to travel the universe. So it is a theoretical paper that this might happen in the future where the data we have right now is the universe is expanding, hence heat death has more support. There is also the big rip which has some supporters. When I say supporters I don't mean they believe one or the other has the data to prove it, just extrapolating from from what we know now seems more likely provided something does not change, which could well occur. But if could be millions or billions of years before we could actually observe these changes to provide support of the big bounce. The straight technical answer is we do not know what the end is, but have some hypothesis of what it could be. Some theories have a bit of data that fits but not enough to say "this is what will happen".

The Big Bounce theory you mentioned is a cyclic universe where things have been bouncing, crunching, bouncing for an infinite about of times. Infinity is big big thing. What I said above is that if you have infinite bounces you will actually get infinite universes like our own, and infinite "you's" exactly like you are now probabilistically speaking. You would also get infinite universes that are not like ours, you would also get infinite universes like ours, but in some of them you don't suffer any anxiety at all and have lived a different life. Might sound contradictory having infinite this and infinite that but the math with infinities are weird, and you can have that. Infinite is really hard to get your head around, when things happen infinitely, things that are of such a ridiculously low probability, such as another bounced universe with the exact copy of you living a copy of your life in this universe become likely, as well and infinite number of universes that can't host life at all etc. etc.

Yes that identical copy of you in a future bounced universe would at a similar universe age (mean this age right now of 13.8 billion years and change) would be posting to reddit and is asking another guy on reddit if "Even if it happens , i won't be the guy , someone new with consciousness will think it's me?" As were the identical copies of you in the past bounces at a same aged universe asked the same question before you came into existence in this universe. In principle you have been doing this exact thing over and over in various bounces where you were identically reproduced bounced universes infinitely many times, while also in other bounced universes have lived other non anxious lives and infinite amount of times (and of course all sorts of other infinite universes with different properties that harbored no life etc.). In those universes identical to this one you are not that guy, it is another guy identical to you, lived the exact same life, thinking the exact some thoughts, but have no connection in any way with the infinite other copies. No connection mentally, or otherwise as each bounce wipes the slate clean, entirely. No evidence of those past "you's" will exist after a bounce. This of course all assumes that the bouncing universe is correct, happening infinitely and does not change like bouncing for a while then doing something else.