Let $S=[(x,y)\in\mathbb{H}^{2}:0< x< 2\pi]$ where $\mathbb{H}^{2}$ is a hyperplane with standard metric. I.e., a strip whose boundary components are geodesics, both approaching a common infinite point.

My simple question is about whether it can be isometrically embedded in $\mathbb{R}^{3}$.

I don't remember exactly but I read some articles about an isometric embedding from a portion of $\mathbb{H}^{2}$ into $\mathbb{R}^{3}$. For example, it might have been proved that infinite polygons of some type can be isometrically embedded but I could not catch the meaning of the type the author said about. Or, I read in another article that every(?) equidistant strip in $\mathbb{H}^{2}$ can be embedded in $\mathbb{R}^{3}$. If this is true, the answer to my question would be positive.

Is there anyone who know about this content precisely?

How about $T=[(x,y)\in\mathbb{H}^{2}:0<\sqrt{{x}^{2}+{y}^{2}}<{e}^{2\pi}]$? This strip also has geodesic boundary components but both components are approaching different infinite point.