Is there a concrete proof showing that hinge loss is an upper bound on the 0-1 loss? While it is stated in several places that the Hinge loss is a convex upper bound on the 0-1 loss, is there a proof behind it? From what I have seen, most resources just show the plots of hinge loss and 0-1 loss and mention that its a convex upper bound on the 0-1 loss. For y>-1 and a small value of y_pred, wont the hinge loss be smaller than 0-1?
Plot of Hinge loss and 0-1 loss

There are several functions called the "hinge loss" and depending on which one you pick, the answer may vary... but from your graph it looks like you're considering $h(x)=\max\{0,1-x\}$. If this is the case, then yes, it is larger than $1_{\mathbb{R}_-}$
Proof:  for any $x\in\mathbb{R}$
\begin{aligned}
h(x)&=
\begin{cases}
1-x &\text{if } x\leq 1\\
0 &\text{if } 1<x
\end{cases}\\
&=
\begin{cases}
1-x &\text{if } x<0\\
1-x &\text{if } 0\leq x\leq 1\\
0 &\text{if }  1 < x
\end{cases}\\
&\geq 
\begin{cases}
1 &\text{if } x<0\\
0 &\text{if } 0\leq x\leq 1\\
0 &\text{if }  1 < x
\end{cases}\\
&=1_{\mathbb{R}_-}
\end{aligned}
