Mixed table borders

You've done it almost the right way. If you want to have a shorter code, you can define a command that will replace all these multicolumns. Here is a way to do it, with some minor improvements to your table:

documentclass{book}
usepackage[utf8]{inputenc}
usepackage[T1]{fontenc}
usepackage{array}
usepackage{xparse}
DeclareExpandableDocumentCommandfcell{O{c}m}{multicolumn{1}{>{$}#1<{$}|}{#2}}

begin{document}

renewcommandarraystretch{1.25}
begin{tabular}{*{8}{>{$}c<{$}}}
    hline
    multicolumn{1}{|c|}{7} & fcell{16} & fcell{3} & fcell{-1} & fcell{9} & fcell{32} & fcell{4} & fcell{2}
    hline
    0 & 1 & 2 & 3 & 4 & 5 & 6 & 7
end{tabular}

end{document} 

You wrote

... The best I could find was using multicolumn, but I guess that's not the proper way to do it, since its real purpose is to merge cells."

Not quite. I'd say that multicolumn has two real purposes: (a) to merge cells -- the purpose you mention -- and (b) to change the column type of its argument, be that a single cell or a range of cells.

To simplify switching between column types, it's frequently handy to set up a shortcut macro. In the example below, the macro mr -- short for "multicolumn-right", I suppose -- is set up for just that purpose.

documentclass{article}
usepackage{array}
newcommand{mr}[1]{multicolumn{1}{r|}{#1}}
begin{document}
$begin{array}{ *{8}{r} }
    hline
    multicolumn{1}{|r|}{7} 
      & mr{16} & mr{3} & mr{-1} & mr{9} & mr{32} & mr{4} & mr{2}
    hline
    0 & 1 & 2 & 3 & 4 & 5 & 6 & 7
    hline
end{array}$
end{document}

With nicematrix.

documentclass{article}
usepackage{nicematrix}
begin{document}
$begin{NiceMatrix}[hvlines,last-row=2]
    7 & 16 & 3 & -1 & 9 & 32 & 4 & 2 
    0 & 1  & 2 & 3  & 4 & 5  & 6 & 7 
end{NiceMatrix}$
end{document}