Shortest distance

Here is an illustration of the laid-out surface area of the cone.

Imgur

We want to find $X$ . We can get it by using the fact that dividing $\theta$ in half will give us two right triangles (as seen in the picture), from which we can use trigonometry to get it.

$S = \sqrt{h^2 + r^2} = \sqrt{5}$

$\theta =\frac{\pi r}{S} = \frac{2\pi}{\sqrt{5}}$

$X = 2 \times (S sin(\frac{\theta}{2})) = 2\sqrt{5}sin(\frac{\pi}{\sqrt{5}})$

$\approx \boxed{4.411}$