That sounds like it would take forever...
The barring gene would need to be introduced from another breed as mentioned
I don't know enough about araucanas and some of the traits such as tufts and rumpless, and blue eggs, etc, and the dominance or recessiveness of those traits, but if they are recessive that is where it would get trickier.
But ignoring all that for a bit, this is how it might be done to get cuckoo into the araucana.
*** warning rookie at work ***
The genetics experts might see a mistake in what I've written here, and if so please feel free to point it out... but I think this illustrates the general approach
1st cross
Black araucana roo x barred Easter Egger hen
results:
all Barred F1 roos (1/2 araucana)
all black F1 hens (1/2 araucana)
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2nd cross
Barred F1 roo x Black pure araucana hen
results:
half the F2 roos are barred, half are black (3/4 araucana)
half the F2 hens are barred, half are black (3/4 araucana)
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3rd cross
Barred F2 roo x Barred F2 hen
results:
all F3 roos are barred, but half are double barred, half are single barred (3/4 araucana)
half the F3 hens are barred, half are black (3/4 araucana)
The F3 Double barred roo has 2 copies of the barring gene and is 3/4 araucana, so this is what you've been working for, so now that you have it you cross it back to a pure araucana
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4th cross
Double Barred F3 roo x Black pure araucana hen
results:
all F4 roos are single barred (7/8 araucana)
all F4 hens are barred (7/8 araucana)
So now you might keeping crossing back to the parent stock of black aruacanas to get more pure araucana traits, or just crossing the F4 offspring together, and selecting the ones that are most true to type (have tufts, blue eggs, rumpless)
Again, this is just my simple example, I'm certain there is more to it due to some of the unique traits of the araucana, but this should give some idea of the process.