Air carrier Captain reported a pressurization system malfunction during arrival. Flight crew continued approach and landed safely.

Narrative

ATC had begun to descend us on the ZZZZZ arrival and had us level at FL270. A few minutes later the Cabin Altitude Horn began sounding and the Red Cabin Alt light was lit up as well. I quickly looked up at the cabin altitude indicator on the pressurization panel and it showed a cabin altitude of approximately 13;000ft. I announced Masks on and we put our masks on and began with our immediate action items followed by the QRC. After coming to the conclusion that the cabin altitude was not controllable; we proceeded with the QRH procedure for Cabin altitude warning or rapid depressurization/Emergency descent.I [requested priority handling] with ATC and they vectored us off the arrival towards lower terrain. Ultimately; we could not get to 10;000ft until we were coming over the front range. I asked ATC to please notify the company of our [situation] and intentions to continue to ZZZ. At some point on the emergency descent; we were able to regain cabin altitude. I was able to get the cabin alt below 10;000ft and we took our masks off and I was able to the communicate with the flight attendants and send Dispatch an ACARS message. I told the FAs (Flight Attendants) we had a pressurization issue and that we had descended to a lower altitude. I told them everything was under control but that we [requested priority handling] and would be on the ground in approximately 8 minutes and to make sure the cabin is prepared for landing and that it would be a normal landing. I made a PA to the passengers explaining we had a pressurization issue and we have descended to a lower altitude and everything was under control and we would be landing shortly.We joined the approach for [Runway] XXL and landed without further incident. ELB (Electronic Logbook) entry was made and I spoke with MX (Maintenance); Dispatch and Chief Pilot per FOM requirements.As far as the fatigue issue; this began on Day 0. I had picked up a XA00 that day. I took a good nap to be properly rested for a duty period that could be 14+hrs. Crew desk called me just over 6hrs into my short call and released me and gave me a XR:20 report the next morning for pairing XXXXXX. I got in bed at XI:30 and wasn't able to fall asleep until sometime after XL:00. I pretty much dozed off and on until my alarm went off around XP:00. Once I got to the hotel in ZZZ1 I grabbed lunch and tried to rest. The next morning; I had a XO:52 van pick up. By the time We got to ZZZ after the pressurization issue I was absolutely beat. I understand the crew desk gave me the proper rest time etc. for my short call ending and new trip starting but there needs to be some sort of procedure for when they essentially are swapping your sleep schedule. I am expected to be rested for a known duty period for my short call duration and I met that. But I can't simply just reset my body to wake up at XP:00 the next morning and expect to be well rested.