Why is it for "Me or her" it was "Ja albo ona", but for this sentence it is "Albo tani albo dobry". Could either sentence structure be used? Like for this one, could it be "Tani albo dobry", and for "Me or her" could it have been "Albo ja albo ona". Or is this sentence like this because it's already a "set" common sentence/phrase?
99,9% of usages of "czy" as "or" are in questions. And also almost every time when you have "or" in questions, it will be "czy".
Then you have the difference between "albo" and "lub". Technically, "albo" should be the 'exclusive or', so it should allow only for one option to be correct. "lub", the 'inclusive or', allows for both options to be correct. In reality, many natives may either simply not know that, or not care.
The exception is this exact construction that we have here. "either... or" shows very strongly that only one option is correct, so it has to be "albo... albo".