How to reduce the number “0” size to fit fraction "1/2"?

I wouldn't reduce the size of the 0 numerals. If you believe that they look too big in relation to the text-style frac{1}{2} expressions, maybe what's really needed is to replace the frac terms with their decimal representations -- of course while aligning the numbers on their (explicit or implicit decimal markers.

documentclass{article}
usepackage{mathtools}  % for 'bmatrix*' env.
usepackage{siunitx}    % for 'S' column type
begin{document}
[
renewcommandarraystretch{1.33}
begin{bmatrix*}[r]
   0           & -frac{1}{2} &  frac{1}{2} 
  -frac{1}{2} &  0           & -frac{1}{2} 
   frac{1}{2} & -frac{1}{2} &  0
end{bmatrix*}
]

[
left[ % note: no need to increase the value of 'arraystretch'
begin{array}{@{} *{3}{S[table-format=-1.1]} @{}}
   0   & -0.5 &  0.5 
  -0.5 &  0   & -0.5
   0.5 & -0.5 &  0
end{array} 
right]
]
end{document}

I would second Mico's recommendation to not do this. But the solution would be to make the fractions display style rather than shrink the zeros (your example was likely set by naïve software which doesn't know how to properly resize fractions. Here's another alternative setting adapted from your MWE:

documentclass{article}
usepackage{amsmath}
usepackage{array}
usepackage{mathtools}
begin{document}
newcommand{half}{{displaystylefrac{1}{2}vphantom{frac{1}{2}^1_1}}}
[
begin{bmatrix*}[r]
  0&  -half &half 
  -half&  0&-half
  half&  -half&0
end{bmatrix*}
]
end{document}

I put the fraction in displaystyle and used a vphantom to add extra spacing—perhaps not enough—above and below the fractions.

The image which the OP posted looks like display style fractions dfrac were used. As stated in my comment, shrinking the 0 in these situations is uncommon. Generally speaking, the various Tex engines and most trusted and well used packages have good typographic features by default.

documentclass{article}
usepackage{amsmath}
usepackage{array}
usepackage{mathtools,xfrac}
begin{document}
renewcommand{arraystretch}{1.2}
newcommand{minus}{scalebox{0.4}[1.0]{$-$}}
[
begin{bmatrix*}[r]
  0&  minusfrac{1}{2} &frac{1}{2} 
  minusfrac{1}{2}&  0&minusfrac{1}{2}
  frac{1}{2}&  minusfrac{1}{2}&0
end{bmatrix*}
]



[
renewcommand{arraystretch}{2.0}%You will see what happens if you do not use it. It is a local definition which does not affect the third matrix
begin{bmatrix*}[r]
  0&  -dfrac{1}{2} &dfrac{1}{2} 
  -dfrac{1}{2}&  0&-dfrac{1}{2}
  dfrac{1}{2}&  -dfrac{1}{2}&0
end{bmatrix*}
]


[
begin{bmatrix*}[r]
  0&  sfrac{-1}{2} &sfrac{1}{2} 
  sfrac{-1}{2}&  0&sfrac{-1}{2}
  sfrac{1}{2}&  sfrac{-1}{2}&0
end{bmatrix*}
]
end{document}

Top: original; middle: dfrac; bottom: https://ctan.org/pkg/xfrac.

I would say that your sample actually suggests the opposite: "I want to preserve regular size even for numerators and denominators". Use dfrac instead of frac.

Also, why do you have special treatment for minus? Why the - isn't enough?

documentclass{article}

usepackage{amsmath}
usepackage{array}
usepackage{mathtools}

begin{document}
renewcommand{arraystretch}{2}
[
  begin{bmatrix*}[r]
    0             &  -dfrac{1}{2} & dfrac{1}{2}  
    -dfrac{1}{2} &  0             & -dfrac{1}{2} 
    dfrac{1}{2}  &  -dfrac{1}{2} & 0
  end{bmatrix*}
]
end{document}

A good solution, in my opinion, to reduce the size discrepancy between fractions (in text mode by default) and ordinary numbers is to use the medium-sized fractions from nccmath, which are ca 80 % of displaystyle:

documentclass{article}
usepackage{nccmath, mathtools} % for 'bmatrix*' env.
usepackage{makecell}

begin{document}

    [ setcellgapes{3pt}makegapedcells
    begin{bmatrix*}[r]
    phantom{-}0 & -mfrac{1}{2} & mfrac{1}{2} 
     -mfrac{1}{2} & 0 & -mfrac{1}{2} 
    mfrac{1}{2} & -mfrac{1}{2} & 0
    end{bmatrix*} 
    ]%

end{document}