Post-determined Array Sorting

Python 2, 67 66 bytes

lambda s,k:`sorted(s,key=lambda c:`k.index(3-ord(c)/32)`+c)`[2::5]

Test it on Ideone.

JavaScript (ES6), 87 bytes

(a,s)=>a.map(n=>[...s].sort().join``.replace([/[^a-z]/g,/[^A-Z]/g,/\D/g][n],``)).join``

If the input array gave the order, rather than the precedence, of the three ranges (this only makes a difference for [1, 2, 0] and [2, 1, 0] whose effects are swapped) then this would have worked for 80 bytes:

(a,s,o=c=>a[(c<'a')+(c<'A')])=>[...s].sort((a,b)=>o(a)-o(b)||(a>b)-(a<b)).join``

I misread the question and still got 7 upvotes with this. Feel free to remove your upvotes and give them to @CharlieWynn instead, who came up with the best correction to my approach.

(a,s)=>a.map(n=>s.replace([/[^a-z]/g,/[^A-Z]/g,/\D/g][n],``)).join``

Pyth, 17 16 15 bytes

s@RSz@L[GrG1`UT

Test suite.

       [          array literal containing...
        G           the alphabet (lowercase)
         rG1        the alphabet, converted to uppercase
            `UT     inspect-range-10, generating the range [0,10) and
                      stringifying it, resulting in a string that contains no
                      letters and all numbers (so equivalent to '0123456789' for
                      filtering)
                    this creates ['ab...z', 'AB...Z', '01...9']

     @L           map over first (implicit) input (the ordering array) and take
                   the nth element in this array for each item
                   this gives us ['01...9', 'ab...z', 'AB...Z']

   Sz             take another line of input as a string and sort it, and then...
 @R               map over intersection: filter the line of input over presence
                    in each element in the new array
                    this results in ['123', 'ac', 'B']

s                 concatenate all and implicitly output

Thanks to @FryAmTheEggman for a byte and @Jakube for another!