Suppose we want to sample $n$ times from the cdf
$$ F_Y= 0.5 + \frac{(a-b)^3}{2 + 2|a-b|^3}$$
where $b$ is the location parameter.
I think it is too difficult to use the inversion transform method, so rejection sampling seems best. So, by differentiating, the p.d.f is given by
$$ f_Y = \frac{-3(a-b)^2(-|b-a|^3+(a-b)^2|b-a|-1)}{2(|b-a|^3+1)^2}$$
I am new to monte carlo sampling, would it suffice to use the proposed distribution as Uniform? Or have I gone about this in the wrong direction?
Inverse transform sampling is very easy to use here. Solve the equation $F_Y=t$ by splitting into two cases: $t\in[0,\tfrac 1 2]$ (so $a\leq b$) and $t\in [\tfrac 1 2,1]$ (so $a\geq b$). In each case $F_Y$ is a fractional linear function and can be explicitly inverted.
Monte Carlo sampling is a broad term - I will discuss rejection sampling. You cannot use the uniform distribution because the support of $Y$ is the whole of $\mathbb R,$ and there is no uniform distribution on $\mathbb R.$ The only constraint is that the proposal distribution pdf $g(x)$ must satisfy $f(x)/g(x)\leq C$ for some $C.$ Since your pdf has polynomial tails ($\sim x^{-4}$), you'd need another distribution with polynomial tails like a two-tailed Pareto distribution.