There should be at least one thread in the archive on this, but unfortunately the search doesn't work for me, so at risk of repeating myself:
Yes, it's the American term meaning give the football to the other team, not the British one meaning to gamble or spend money.
In American football, the team that has the ball has four chances/tries/attempts ('downs') to advance the ball 10 yards toward the opposite goal. If they don't get the 10 yards, they lose possession: the ball is given to the other team, who in turn will try to advance it the other direction.
So, if after the first three downs (tries, attempts) they have not made enough progress to have a reasonable hope of completing the 10 yards and keeping the ball, the prudent play on the 4th down is to kick (= punt) the ball far away down the field, so that when the opposing team receives possession, they will not be in the middle of the field where the teams are playing, but much farther from the goal.
A rough equivalent in soccer is a long, high defensive kick far down the field, which does give up the ball to the other team, but places it so far away that they have to start all over with a new attack.
So the point with regard to Syria is that there is no positive offensive strategy; the 'punt' option is a defensive, last-choice option, one that cedes territory and momentum, but mitigates loss and (hopefully) prevents catastrophe.
Figuratively, 'punt' typically implies something like 'give up' or 'shrug one's shoulders and admit defeat.' It's not a desirable or admirable choice, only one borne of necessity.