Answer: Approximately \(0.251h\).

Differentiate \(y=\sin x+e^{-x}\):

\(\frac{dy}{dx}=\cos x-e^{-x}.\)

For a small increase \(h\),

\(\Delta y\approx h\frac{dy}{dx}.\)

At \(x=\frac{\pi}{4}\),

\(\frac{dy}{dx}=\cos\frac{\pi}{4}-e^{-\pi/4}.\)

Thus

\(\Delta y\approx h\left(\cos\frac{\pi}{4}-e^{-\pi/4}\right).\)

Numerically,

\(\cos\frac{\pi}{4}-e^{-\pi/4}=0.251\ldots.\)

So the approximate change in \(y\) is

\(\boxed{0.251h}.\)