The first thing to note is that some standard sudoku principles apply. For example for each row and column as well as each block can only contain a particular digit 3 times in each place.

I will use notation [row,column].

To start off we can notice that in the fifth row all 3 of the right zeros ( -0) are filled in so entry [5,4] (2-) must be equal to 22 since the block containing it has a 21.

Then entry [4,2] must be 22 as the row and column of it contains the others.
Entry [4,9] must be 12.
Entry [3,5] must be 20 because of the column.
Entry [3,1] must be 22.
Entry [3,6] must be 01 as the other -1 entries are filled.
Entry [1,8] must be 10 by the 1- entries in the block.
Entry [5,8] must be 20 by the column.
Entry [6,3] must be 12 by column and block.
Then [6,1] must be 02 by elimination.
Entry [7,1] must be 12 by column.
Entry [1,4] must be 00 since row below contains 00 and block contains 20.
Entry [1,5] must be 12 by row and column.
Entry [5,2] must be 01 by block and column.
Entry [9,5] must be 11 by column and row.
Then [9,3] must be 21 by elimination.
Entry [7,4] must be 10 by block and column.
Entry [8,6] must be 21 by block and column.
Entry [6,5] must be 00 by row and column.
Entry [2,5] must be 10 because adjacent columns already contain it and the block must have one.
Entry [3,3] must have one by a similar argument.
Entry [3,9] must be 00 because rows above contain it and adjacent column contains it and the block must have one.
Then this implies that [5,9] below must be 10.
By the same token [5,7] must be 00.

At this point we can actually convert the whole puzzle to a standard Sudoku puzzle and solve it by known means.

To convert we use Dec(Trinary) + 1. So if we have a 2 digit trinary number ab then Dec(ab) = 3*a + b in decimal. We add the one to have the numbers span 1-9 instead of 0-8.

Or using the following key:

! 00 - 1 01 - 2 02 - 3 10 - 4 11 - 5 12 - 6 20 - 7 21 - 8 22 - 9

So we get the regular Sudoku puzzle:

! [5,X,3][1,6,X][X,4,X]
[X,1,X][X,4,X][6,X,8]
[9,X,4][8,7,2][5,X,1]

[X,9,X][X,8,4][3,X,6]
[X,2,X][9,X,X][1,7,4]
[3,4,6][X,1,X][X,8,X]

[6,X,X][4,9,X][X,1,X]
[X,7,X][X,X,8][X,5,X]
[X,X,8][X,5,X][2,X,X]

Which has the solution

! [5,8,3][1,6,9][7,4,2]
[2,1,7][3,4,5][6,9,8]
[9,6,4][8,7,2][5,3,1]

[7,9,1][5,8,4][3,2,6]
[8,2,5][9,3,6][1,7,4]
[3,4,6][2,1,7][9,8,5]

[6,5,2][4,9,3][8,1,7]
[1,7,9][6,2,8][4,5,3]
[4,3,8][7,5,1][2,6,9]