We assume that in addition to the point $z$, we also are given the real axis, and a unit distance. So we can draw the unit circle.
If $z\ne 0$, then $\dfrac{1}{z}$ has norm the reciprocal of the norm of $z$. Its argument is the negative of the argument of $z$. So if we find the number $w$ on the ray through the origin and $z$ such that $w$ has norm the reciprocal of the norm of $z$, and reflect in the real axis, we will have constructed $\dfrac{1}{z}$. Reflection is easy using the Euclidean tools. (Thanks to Phira for pointing out an earlier error.)
This brings us to a classical problem of Euclidean geometry: Given a length $a\gt 0$, draw a line segment of length $\dfrac{1}{a}$.
Draw a pair of axes, which for convenience we call the $u$ and $v$ axes, meeting at some point $O$. (The axes do not need to be perpendicular.)
On the $u$-axis, draw a point $A$ such that $OA=a$, and a point $U$ such that $OU=1$. On the $v$-axis, draw a point $V$ such that $OV=1$. Join $AV$. Through $U$, draw a line parallel to $AV$. Suppose that this line meets the $v$-axis at $B$.
Because $\triangle OAV$ and $\triangle OUB$ are similar, we have $\dfrac{OB}{OU}=\dfrac{OV}{OA}$, and therefore $OB=\dfrac{1}{OA}=\dfrac{1}{a}$.
All the constructions described above can be done with compass and straightedge.