I have been since nearly the beginning.
Refers to the biblical Eve
I endure nearly to the end of all time.
At first, I was the one that came after.
Again, the biblical Eve, who came after Adam
Now, I am the one that comes before.
Eve as in the day before, like today (X-mas eve, which also explains the title)
I am −1.
If today is $0$, then eve is $-1$ (alternatively, the date of the eve is one less than the date of the actual day)
I am 3 of 7.
Three letters of sEVEn
I am sometimes confused for 5 or 22.
EVE can be anagrammed to VEE (V, five in Roman numerals, or the 22nd letter)
If n is an integer, I am 2n−1.
$2n$ is even, so $2n-1$ is EVE(-n)
I might be cut off the middle.
sEVEr (thanks @Ankoganit)
I might be part of an event, or merely part of the countdown.
EVEnt, or part of the countdown to the following day (thanks @Rubio)
I might be found in a lost flight of birds.
A flight of birds forms a VEE-shape, which anagrams (lost, in the confused sense) to EVE (thanks @Ankoganit)
I might be the first and last thing you ever achieve.