Answer: \(3\pi\left(\frac{1000}{\pi}\right)^{2/3}\approx 439\text{ cm}^2\).

The volume is

\(\pi r^2h=1000.\)

So

\(h=\frac{1000}{\pi r^2}.\)

Since the cylinder is open, its outer surface area is

\(S=\pi r^2+2\pi rh.\)

Substitute \(h=\frac{1000}{\pi r^2}\):

\(S=\pi r^2+2\pi r\left(\frac{1000}{\pi r^2}\right).\)

Hence

\(S=\pi r^2+\frac{2000}{r}.\)

Differentiate:

\(\frac{dS}{dr}=2\pi r-\frac{2000}{r^2}.\)

For a stationary value,

\(2\pi r-\frac{2000}{r^2}=0.\)

Thus

\(2\pi r^3=2000,\)

so

\(r^3=\frac{1000}{\pi}.\)

Therefore

\(r=\left(\frac{1000}{\pi}\right)^{1/3}.\)

At this value, \(\frac{2000}{r}=2\pi r^2\), so the minimum surface area is

\(S=\pi r^2+2\pi r^2=3\pi r^2.\)

Hence

\(S_{\min}=3\pi\left(\frac{1000}{\pi}\right)^{2/3}.\)

Numerically,

\(\boxed{S_{\min}\approx 439\text{ cm}^2}.\)