If I'm reading your definition correctly, this category looks equivalent to the category ${\bf Set_*}$ of pointed sets and basepoint-preserving functions (the equivalence is by removing the basepoint from each pointed set: you're left with an ordinary set and possibly partial functions). So you should be able to take the ordinary product in ${\bf Set_*}$ and then passing it through the equivalence: the product of $X$ and $Y$ in ${\bf pSet}$ should be the disjoint union of the cartesian products $X\times Y$, $X\times\{*\}$, and $Y\times\{*\}$, whose projections to $X$ and $Y$ are the projections from $X\times Y$ and undefined on the other two pieces.

If I'm reading your definition correctly, this category looks equivalent to the category ${\bf Set_*}$ of pointed sets and basepoint-preserving functions (the equivalence is by removing the basepoint from each pointed set: you're left with an ordinary set and possibly partial functions). So you should be able to take the ordinary product in ${\bf Set_*}$ and then pass it through the equivalence: the product of $X$ and $Y$ in ${\bf pSet}$ should be the disjoint union of the cartesian products $X\times Y$, $X\times\{*\}$, and $Y\times\{*\}$. The partial projection to $X$ is given by projection from $X\times Y$ and $X\times\{*\}$ and undefined on $Y\times\{*\}$, and the partial projection to $Y$ is similar.

And indeed, this works: if $C$ has partial functions $f$ and $g$ to $X$ and $Y$ respectively, then we get a partial function to $(X\times Y)\sqcup (X\times\{*\})\sqcup (Y\times\{*\})$ given by $c\mapsto (f(c),g(c))$ if both exist, $(f(c),*)$ or $(*,g(c))$ if only one does, and undefined if neither exists.