You can essentially use the same approach as in your MWE except that you use figure to number your theorems:
\numberwithin{figure}{section}% number figures inside sections
\newtheorem{teo}[figure]{Theorem}
\newtheorem{ex}[figure]{Example}
The \numberwithin command, which comes from amsmath, numbers the figures in tandem with the sections. Using this approach you can get the output

using the code:
\documentclass{amsart}
\numberwithin{figure}{section}
\newtheorem{teo}[figure]{Theorem}
\newtheorem{ex}[figure]{Example}
\begin{document}
\section{Let the section begin}
\begin{teo}The sum of two numbers is a number\label{T:one}\end{teo}
\begin{figure}[h]A figure\caption{Nice one}\label{F:one}\end{figure}
\begin{ex}For example, $1+2=3$.\label{E:one}\end{ex}
See \ref{T:one}, \ref{F:one} and \ref{E:one}.
\end{document}