Air carrier First Officer reported a NMAC while on approach. The First Officer followed the procedures for an RA event.

Narrative

We were flying the ZZZZZ 3 RNAV Arrival into ZZZ. At some point during the arrival; we received vectors to Runway XXR. We had extended the approach course off of ZZZZZ1 and called the field in sight. ATC cleared us on a visual for Runway XXR. We had already reviewed the criteria to go to TA only on the approach; which are as follows.Place TCAS mode selector to TA when all of the following criteria are met: 1. Cleared for any approach to Runways XXL or XXR 2. Notified of any conflicting traffic 3. Visual separation can be maintained throughout the approach. The only criteria we met as we began the approach was Number 1. We intercepted the course in VOR/LOC; selected APP mode and when we intercepted G/S zeroed out the altitude window. We intercepted the final around 8000 ft. MSL just outside of ZZZZZ1. While we were on glideslope and configuring; we received a TA. I noted the TA was behind us. Seconds later we got a call from the Tower about a conflicting aircraft. I placed my hand on the TA/RA switch and immediately looked for the traffic. I could not see it because he was behind us. So; at that point; we had met criteria 1 and 2; but not 3. We then got an RA alert; I took my hand off the TA/RA switch. I looked back in the cockpit and saw the conflicting aircraft was within 500 ft. and appeared to be closing. The First Officer who was flying immediately responded to the RA; which was a momentary climb; and then leveled off. I notified ZZZ Tower we were responding to an RA. We had been on autopilot and auto throttle at the time; which the First Officer immediately disengaged to respond to the RA. I do not recall ever hearing the automation disengage warnings because I was so tuned to the RA and had turned again to look for the (Other carrier) aircraft. This time I saw him belly up to us pulling away and down to reintercept the [Runway] XXL final approach course. I commented to the FO that the other aircraft was very close. The PF (Pilot Flying) asked if we were safe to continue or if we should go-around. I could visually confirm we were clear of the conflict and asked if TCAS also showed us 'clear of conflict'. She confirmed we were and since we were only a dot above glideslope and gear down with Flaps 15; I elected to continue and assess our situation at 1;000 ft.By this point; (Other carrier) had passed ahead of us; was in the process of regaining final and landing before we did on Runway XXL. We continued the approach and were on speed; on descent profile; and configured at 1000 ft. so we continued the approach uneventfully. When we cleared the runway the First Officer remarked to Tower with something to the effect of; 'that was really close.' Tower responded with something like; 'yeah; closer than comfortable.' Tower then asked if we wanted a number to call. I said 'no'. At the time I thought the closest the other aircraft had come to; was approximately 500 ft. However; the First Officer who was concentrating on the instruments and the RA procedure and could not see the aircraft because it was on my side of the aircraft; noted that the aircraft had come to within 100 ft. As we taxied in; we sent a quick ACARs message to Dispatch; who responded with a request to call [Person A] once we parked. After we finished deboarding and handing over the aircraft to the next Crew; I called Dispatch and informed them of the event.I don't believe ATC or us could have done anything different. The aircraft that created the conflict was behind us at the time and clearly overshot the final approach course. They also had enough speed to fly past us and land before us. Tower warned us of the conflict just after we had received a TA. Once we got the RA; we followed the TCAS procedures and reported the event. I believe any and all preventive measures can only be addressed by the other aircraft.