The minimal Sudoku contains 17 digits: 2 each of seven of the nine digits, 3 of another digit, and none of the last one. (By symmetry, it doesn't matter which two digits are selected to appear 3 and 0 times in the Sudoku.)
If we select our two primes as
2 and 3, then we can represent the digits as $$3-2,2,3,2+2,2+3,3!,?,2\wedge3,3\times3.$$ (We omit 7 because we only need to represent all but one of the digits for the minimal Sudoku.)
So we choose the minimal Sudoku with
seventeen digits filled, two each of $1,2,4,5,6,8,9$ and three of $3$.
How many symbols do we need to use then?
Each of $1,4,5,8,9$ requires three symbols, $2$ and $3$ require only one, and $6$ requires two. So the total is $$5(2\times3)+(2\times1)+(2\times2)+(3\times1)=30+2+4+3=39.$$