Solution:
Let $C$ be the original cactus. For every cycle in $C$, arbitrarily remove one of its edges, yielding a new graph $T$. Observe that since the cycles are edge-disjoint, we removed exactly one edge from every cycle, meaning the graph stays connected. However, there are no longer any cycles, so $T$ is a tree.

Now consider any leaf $v$ of $T$ (note that any nonempty tree must have a leaf). If $v$ is the only vertex in the graph we're trivially done, since then $v$ was the only vertex in $C$. Otherwise $v$ has degree $1$. If $v$ was originally a leaf of $C$, we're done. If not, observe that in the process of turning $C$ into $T$, a vertex's degree cannot decrease by more than half, because for every cycle that a vertex is part of in $C$, it gains a degree of $2$, but can only lose $1$ degree from an edge of that cycle being removed. Therefore, the original degree of $v$ in $C$ was at most $2$, meaning it could have been part of at most $1$ cycle, as desired.