sudoku - This is a crossudoku

So, this is a crossudoku.

It is the same as a sudoku - same rules apply. However, you have been given crossword-like clues for part of the numbers on the grid.

Some of the clues may have multiple answers - so you may need to be solving the "crossnumber" and the "sudoku" part of the puzzle simultaneously to find the correct digit for each cell.

A1-C1 and C9-E9 Then will I swear beauty herself is black, And all they foul that thy complexion lack.

A2-B2, I5-I6, I7-I9 and B6-B8 A perfect square and a perfect cube

A5-G5 A quick and easy approximation of a pie

E3-F3 The answer.

F2-G2 and G4-H4 The number of months in a year

G6-H6, G7-H7, C6-D6 and B9-C9 The sum of two consecutive squares

D4-E4, H1-H2 and I6-I7 There is a lot of silver lining hidden in this grid...

E8-H8 My kingdom is not of this world.

C7-D7 The sum of two consecutive cubes

A4-A7 The devil was hit by a cube

B5-B8 Your ride when you are drunk

I8-I9 The sum of 3 consecutive squares

I3-I4 Some buildings skip this floor

F3-F4 and G2-G3 Between a square and a cube

Answer

The filled grid looks like this:

Several of the clues have only a single possibility considering that we know the number of digits and that all digits have to be unique. The second clue especially gives a little of information. Starting with around half of the clues entered into the grid and solving with the sudoku rules we can slowly decipher the other clues and solve the whole grid.

The solution to the clues are:

A1-C1 and C9-E9 Then will I swear beauty herself is black, And all they foul that thy complexion lack.

132 (Sonnet 132 by Shakespeare)

A2-B2, I5-I6, I7-I9 and B6-B8 A perfect square and a perfect cube

64, 64, 729, 729 (64 and 729 are the only perfect squares and cubes with 2 or 3 digits)

A5-G5 A quick and easy approximation of a pie

3142857 (22/7 is a approximation of pi, whose first digits are 3.142857)

E3-F3 The answer.

42 (The answer to everything)

F2-G2 and G4-H4 The number of months in a year

12

G6-H6, G7-H7, C6-D6 and B9-C9 The sum of two consecutive squares

85, 41, 61, 61 (Possibilities are 13, 25, 41, 61, 85)

D4-E4, H1-H2 and I6-I7 There is a lot of silver lining hidden in this grid...

47 (Atomic number of silver (thanks Lolgast))

E8-H8 My kingdom is not of this world.

1836 (Quote from John 18:36 (thanks Lolgast))

C7-D7 The sum of two consecutive cubes

35 (35, 91 are the possibilities)

A4-A7 The devil was hit by a cube

5328 (8 * 666)

B5-B8 Your ride when you are drunk

1729 (Taxicab numbers)

I8-I9 The sum of 3 consecutive squares

29 (14, 29, 50, 77 are the possibilities; the clue is not needed because of the second clue)

I3-I4 Some buildings skip this floor

13 (Some buildings skip the 13th floor because of superstition)

F3-F4 and G2-G3 Between a square and a cube

26 (Lies between 25 and 27, the only number to do so with 2 digits)