B767 flight crew reported receiving a low altitude alert from ATC during approach.

Narrative

On arrival from the ZZZZZ arrival to the ILS XX into ZZZ we were inside of ZZZZZ1 and were given a frequency change to another Approach Control frequency. We had either copied the frequency wrong or given the wrong frequency and told to return to the last frequency. This happened twice and while were working this out we had to remain at our last cleared altitude of 6000. Once we got the frequency issue worked out we were high and needed to descend to the approach intercept altitude. Our clearance was to maintain 210 kts. and above 2000 until established and cleared for the ILS XX approach. As we had already intercepted the Localizer and were inside the IAF I armed the approach mode and commanded the FO to set 1500 in the altitude window of the Mode Control Panel (MCP); the final approach fix altitude and the MCP to flight level change 210 kts trying to catch up on our higher than normal intercept on the approach. At that time I was expecting the aircraft to level off at 2000 until intercepting the glide slope as it was the next altitude at the ZZZZZ2 fix on the approach. The aircraft proceeded to descend to 1500 the set final approach fix altitude without leveling at the 2000 ZZZZZ2 altitude. I didn't catch that mistake and believed we were okay to descend to the 1500 FAF altitude to intercept the glide slope. Approaching 1600 msl (400 low) we received a call from the Approach Controller stating that we got a low altitude alert as we were below the minimum altitude for ZZZZZ2 and to climb back to 2000. We did as requested and were subsequently re-cleared for the approach and landed without incident.