“One of its variants”? You mean Al-Kuhen, Caen, Cahan, Cahen, Cahn, Cahon, Cahona, Cain, Choen, Coen, Cofen, Coffen, Cohan, Cohane, Cohen, Cohn, Cohne, Cone, Conn, Coon, Cowan, Cowen, El-Kohen, Kagan, Kaganovich, Kahana, Kahane, Kahano, Kahanow, Kahansky, Kahin, Kahn, Kan, Kaplan, Katz, Kogan, Koganovitch, Kogen, Kogon, Kohenteb, Kohentov, Kohn, Kohne, Kohner, Kohnowsky, Kohányi, Koihen, Kon, Konstamm, Koon, Kouihen, or Xohen ?
Here’s the clading tree for J-P58; as you can see the P58 mrca was 6500 BC or so. Most of J1 lies in the P58 clade. A very large portion of Cohens fall in the Y3088 subclade of P58, about ⅛ of the way down the above page. It’s mrca is 900 BC ±500, about what might be expected for an ancient Israeli priest progenitor.
As shown here there are other haplogroups associated with the Cohen surname. (This page might be the easiest Google hit for Cohen Y-haplogroups, but doesn’t show well the large cluster at Y3088/Z18271.)