Dirichlet's proof is described in <A HREF="https://books.google.nl/books?id=qEwpwWyVPIAC">Number Theory:  Algebraic Numbers and Functions</A> (starting on page 48).

Dirichlet  did  not  use  Minkowski’s  theorem;  he  proved  the  unit  theorem  in  1846 while Minkowski’s theorem appeared in 1889.  Dirichlet’s substitute for the convex-body theorem was the pigeonhole principle.  (An account of Dirichlet’s proof in German is in [2, Sect.  183]and in English is in [6,  Sect.  2.8–2.10].)  Dirichlet did not state the unit theorem for all orders, but only those of the form $\mathbf{Z}[\alpha]$, since at the time these were the kinds of rings that were considered.

<A HREF="https://home.mathematik.uni-freiburg.de/arithgeom/lehre/ss20/algzt/Keith%20Conrad%20-%20Unit%20theorem.pdf">[source]</A>