Air carrier Captain reported during initial approach receiving a TCAS alert of an aircraft that was 400 feet above them. Captain took evasive action.

Narrative

Coming into BUF; we were being vectored for the instrument approach ILS 32. Conditions were clear. Our final vector occurred at approximately 10-12 miles away from the airport. ATC assigned a heading and a descent from 4;000' to 3;500'. As we turned to join the localizer; a TCAS caution flagged. He was 400' above us; in the direction of our turn and easy to spot. It became immediately obvious that if we continued our turn; we would be coming unreasonably close to the conflicting traffic. I turned off the autopilot and began a turn in the opposite direction. Simultaneously; we received the warning with a 'monitor vertical rate' alert. Admittedly; I was outside; fixated on the traffic; making sure we didn't come any closer. I began a shallow climb of 200 fpm. My FO called out the error and I corrected. When we alerted ATC; he seemed casual and confident that we should have been below the traffic. The order of events occurred in seconds. Two things stood out to me. First; ATC should have gave us more time. Best case scenario; we would have gotten a TCAS caution before implementing VS to descend. Second; the warning came way later than helpful. The turn obviously exacerbated the issue. However; the two of us didn't hesitate for a second. We saw the issue before the plane issued a warning and responded in a way that kept everyone safe. As we deplaned; the flight attendants were giggling with the passengers.Cause: VFR traffic was either low or had a transponder transmitting 100' low. ATC put us right under the guy.