im a level 9 punisher.
and to be honest im not so familiar with the skills and how to use them.
Actually I had trouble when I did this quest with my Punisher too. No problems at all on Whipper but yeah, Punisher for some reason I couldn't get the damage out fast enough.
What I know I haven't done is build the base skills properly. I'm a little cautious of maxing out lower lvl skills and then not having access to the higher ranked skills around lvl 26 or so due to lack of skill points (has happened to me on other games).
Anyway, as people have mentioned, start of the fight drinking a HP pot, drink another as soon as the cooldown finishes and you should be ok. Also, start the fight from around 25M away or so. Once you select the boss and hit one of your skills, you'll automatically move within range to use the skill. It should give you maybe 2-3 hits before he gets in range of you.