Finite Cantor's Diagonal

CJam (15 14 bytes)

qN/ee{:=i2%)}%

Online demo

Thanks to Adnan for a one-byte saving.

Dissection

qN/    e# Split input on newlines
ee{    e# Label each line with its index and map:
  :=i  e#   Get the character at the same index (i.e. diagonalise)
  2%)  e#   Compute (ASCII value mod 2) + 1
       e#   Maps 0 1 2 3 4 5 6 7 8 9
       e#     to 1 2 1 2 1 2 1 2 1 2
}%

Python 2, 47 45 bytes

lambda x,n:int(`x`[1::n+3])%(10**n/2)+10**n/9

Thanks to @xnor for golfing off 2 bytes!

Test it on Ideone.

How it works

`x` yields a string representation of the list x.

For the first test case, this gives the string

[92345678, 23456789, 34567890, 45678901, 56789012, 67890123, 78901234, 89012345]

[1::n+3] retrieves every (n + 3)^th character – where n is the length of x starting with the second one. Accounting 2 characters for ,, we retrieve the first digit of the first number, the second digit of the second number, etc.

[92345678, 23456789, 34567890, 45678901, 56789012, 67890123, 78901234, 89012345]
 ^          ^          ^          ^          ^          ^          ^          ^

We now take the number modulo 10ⁿ ÷ 2 to map the first digit in the range [0, 4].

For 93579135, we get 93579135 % 50000000 = 43579135.

Finally, we add 10ⁿ ÷ 9 to the last result, which increments – wrapping around from 9 to 0 – all digits by 1 (no carry) or 2 (with carry).

For 43579135, we get 43579135 + 11111111 = 54690246.

MATL, 11 10 9 bytes

VXd9\QV!U

Takes only a column vector of integers as input. N is not provided.

Try it Online

Explanation

    % Implicity grab input as column vector of numbers
V   % Convert the input column vector into a 2D character array
Xd  % Grab the diagonal elements of the character array
9\  % Take the modulus of each ASCII code and 9
Q   % Add 1 to remove all zeros
V   % Convert the result to a string
!   % Transpose to yield a row vector of characters
U   % Convert back to an integer (as per the rules)
    % Implicitly display result