Can I ignore the Nyquist criterion? If I have a signal with frequencies up to 2000 Hz but I'm only interested in the frequency range up to 500 Hz, is it correct to sample at 1000 Hz $(f_\mathrm s \geq 2 \cdot f_\mathrm{max})$? Or do I have to apply an analog lowpass filter with cut-off frequency 500 Hz first?

I read this similar post, but didn't understand it well:
Ignoring Nyquist
You can only "ignore Nyquist" if you a-priori know that there is no higher frequency content (above your desired or required noise floor) that will alias down (or up!) to your band of interest.  But since your signal has content at and above half your sample rate, you will need to low pass filter that signal before sampling.

You will also need a sampling rate ABOVE 1000 Hz, if you are interested in a finite length of signal at 500 Hz.