This is a truncated Gaussian density: $$\mathcal{R}(x|\mu,\sigma,\alpha)\propto K\exp\left(-\frac{\left(x-\left(\mu-\sigma^2\alpha\right)\right)^2}{2\sigma^2}\right)\mathbb{I}_{x>0}$$ means that the posterior distribution is proportional to a truncated Gaussian density but the proportionality factor $K$ is defined by the fact that it is a density! Hence it is the very same truncated Gaussian density.

With respect to the overall question, there are several ways of sampling from a truncated Gaussian, the easiest one being by using the inverse cdf transform. I also wrote an accept-reject algorithm in parallel with John Geweke about twenty years ago.)

This is a truncated Gaussian density: $$\mathcal{R}(x|\mu,\sigma,\alpha)\propto K\exp\left(-\frac{\left(x-\left(\mu-\sigma^2\alpha\right)\right)^2}{2\sigma^2}\right)\mathbb{I}_{x>0}$$ means that the posterior distribution is proportional to a truncated Gaussian density but the proportionality factor $K$ is defined by the fact that it is a density! Hence it is the very same truncated Gaussian density.

With respect to the overall question, there are several ways of sampling from a truncated Gaussian, the easiest one being by using the inverse cdf transform. I also wrote an accept-reject algorithm in parallel with John Geweke about twenty years ago.

This is a truncated Gaussian density: $$\mathcal{R}(x|\mu,\sigma,\alpha)\propto K\exp\left(-\frac{\left(x-\left(\mu-\sigma^2\alpha\right)\right)^2}{2\sigma^2}\right)\mathbb{I}_{x>0}$$ means that the posterior distribution is proportional to a truncated Gaussian density but the proportionality factor $K$ is defined by the fact that it is a density! Hence it is the very same truncated Gaussian density.

This is a truncated Gaussian density: $$\mathcal{R}(x|\mu,\sigma,\alpha)\propto K\exp\left(-\frac{\left(x-\left(\mu-\sigma^2\alpha\right)\right)^2}{2\sigma^2}\right)\mathbb{I}_{x>0}$$ means that the posterior distribution is proportional to a truncated Gaussian density but the proportionality factor $K$ is defined by the fact that it is a density! Hence it is the very same truncated Gaussian density.

With respect to the overall question, there are several ways of sampling from a truncated Gaussian, the easiest one being by using the inverse cdf transform. I also wrote an accept-reject algorithm in parallel with John Geweke about twenty years ago.)