I missed the train because I was mistaken about the time of my train. In this case, should I say, "I got mixed up with the times." or "I got the times mixed up. Another example is that I mistook X from Y. They are two people. In this case, which sentence is correct, "I got mixed up about X and Y." or "I got X mixed up with Y."?
Get A and B mixed up means to confuse A with B. It requires at least two terms, whose identities you transpose in some important respect, thinking that A is B and B is A. For instance
I get Abbott and Costello mixed up.
This means you can't remember which is the short one and which the tall one.
In your second example, where you mistook X for Y, you want to use "I got X and Y mixed up".
Get mixed up about X means to be in a state of confusion about X. It requires only one term. You may say that you get mixed up about A and B, but in this case "A and B" is a single topic involving both A and B.
I get mixed up about Abbott and Costello.
This means that there is something you don't understand about the famous comedy team—perhaps whether they came before or after Gallagher and Shean.
In the case of "I missed the train because I was mistaken about the time of my train", there is only time involved, so you want to say "I got mixed up about".
Both versions are acceptable in casual speech, but note that both got and mix up are relatively informal usages here. More "formal" versions here would be, for example,...
1: I was confused about X and Y (being faced with X and Y made me confused)
2: I confused X and [or with] Y (I failed to distinguish X from Y, and vice-versa)
...from which it may be easier to see that #1 doesn't quite say the same thing. If, for example X and Y were complex concepts, maybe you simply didn't understand one or both of them.
In practice, if X and Y were simple things (such as the departure time of your intended train, and another one going in the same direction that didn't actually stop at your destination station), then people would understand the second meaning even if you used the first form. But to be precise, just use #2 anyway.