Let $T_B=\left[\begin{array}{ll}A& B\\ 0& D\end{array}\right]$ be an unbounded upper operator matrix with diagonal domain, acting in $\mathcal{H}\oplus \mathcal{K}$, where $\mathcal{H}$ and $\mathcal{K}$ are Hilbert spaces. In this paper, some sufficient and necessary conditions are characterized under which $T_B$ is Browder (resp., invertible) for some closable operator $B$ with $\mathcal{D}(B)\supset \mathcal{D}(D)$. Further, a sufficient and necessary condition is given under which the Browder spectrum (resp., spectrum) of $T_B$ coincides with the union of the Browder spectrum (resp., spectrum) of its diagonal entries.