Derivative of argmin in a constrained problem
Entire books have been written about this subject. See for example:
Bonnans, J. Frédéric, and Alexander Shapiro. Perturbation analysis of optimization problems. Springer Science & Business Media, 2013.
Fiacco, A. V. Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press New York. 1983.
The basic idea here is that under conditions of smoothness and constraint qualification, for fixed $y$, any minimum of the constrained optimization problem will satisfy the KKT conditions, which are a system of $m+n$ nonlinear equations in $m+n$ unknowns ($n$ for the variables $x$, and $m$ for the Lagrange multipliers.) If the solution to the KKT conditions is isolated, and the perturbations of the constraints are smooth, then you can apply the implicit function theorem to the solution of the KKT conditions to show that the optimal $x^{*}$ and $\lambda^{*}$ are locally smooth functions of $y$. You can then solve a linear system of $m+n$ equations in $m+n$ unknowns to find the derivatives with respect to a change in $y_{i}$.
However, if the solution to the KKT conditions is not isolated (e.g. if there are multiple sets of Lagrange multipliers that work), then the derivatives that you seek may not exist.
The references that I've given above state rigorous theorems. We would have to know more about your particular problem to determine whether the theory is applicable to your problems.
The particular case of sensitivity analysis for parameterized linear programming problems is also well understood and you can find discussions of it in many textbooks on LP.
The specific case of $x$ and $y$ both being one-dimensional real variables is simpler than the multidimensional picture. In particular, there is no need for Lagrange multiplier, as it is often easiest to reason without them. (Lagrange multipliers are formally valid in 1D but useless. If you want more details on why this is so, please ask a separate question unless there is one already.)
Let $h_b$ be the function you defined, now making the parameter $b$ explicit.
The argument I gave in response to an earlier question shows that if $f$ is twice continuously differentiable, the minimum is unique, and $\partial_1^2f(h_\infty(y),y)>0$ for all $y$, then $h_\infty$ is continuously differentiable with $$ h_\infty'(y) = -\frac{\partial_2\partial_1f(h_\infty(y),y)}{\partial_1^2f(h_\infty(y),y)}. $$ This was for the unconstrained problem.
If $h_\infty(y)<b$, then $h_\infty(y)=h_b(y)$. By continuity if this is valid at a point $y$, it will be valid in a neighborhood. In this case $h_\infty'(y)=h_b'(y)$. If the minimizer is in the interior of the domain, it all behaves as if there was no constraint. This is a typical feature.
However, bear in mind that the formula for the unconstrained case required a twice continuously differentiable function to start with. With only one derivative I foresee trouble, and things can go awry.
If you are at the boundary (again assuming a global minimum for all $y$), then there are two possible cases:
$\partial_1f(h_b(y),y)<0$
$\partial_1f(h_b(y),y)=0$
The case with $>$ is impossible, as then you could never be at a minimum at the endpoint of the interval. In case 1 the derivative of the condition will continue to be negative in a small neighborhood, and the minimizer will stay at the boundary when you vary $y$ a little. Thus $h_b(y)=b$ for $y$ in some open interval and so $h_b'=0$ at the point.
In case 2 differentiability can fail. Take, for example, $f(x,y)=(x-y)^2$, for which $$ h_b(y) = \begin{cases} y, & y \leq b\\ b, & y \geq b. \end{cases} $$ This function is continuous but not differentiable across the point $y=b$ where you have the "transition type" between interior minimizers and stable boundary minimizers (case 1).