How is the redshift of starlight precisely measured? When galaxies move away from us (caused by the expanding spacetime) their light seems to show a redshift. But what is really needed for this?

What time lap is for example needed to see a difference in frequency, and how does this precisely work?

Is this also possible for gravitational waves, or are amplitudes the only solution to derive the redshift from?
I don't really think I fully understand the first part of your question.

Redshift is measured by comparing the wavelengths of a redshifted pattern of absorption or emission lines with the wavelengths they would have in an object at rest. All lines are shifted by the same factor of $1+z$, where $z$ is the redshift.

Gravitational waves cannot be used to obtain a redshift because the intrinsic wavelength of the waves is not known. An inspiralling binary emits waves with a frequency equal to twice the orbital frequency. This in turn depends on the mass, but the observed frequency is also redshifted. There is thus a "degeneracy" between the mass and redshift.

What you can do with gravitational waves is estimate the physical distance to the source using the wave amplitude and how quickly the amplitude changes with time. If the host galaxy of the source can then be identified, then you can use the redshift of that galaxy to get an estimate of the Hubble parameter (e.g. see Abbott et al. 2017).