So its one or the other? I thought you would (in this example) you would reduce your weapon value by one and then after every coming action round (after you move the ship) the person tries to roll a hit or a critical hit, if he rolls neither the weapon value continues the same as stated earlier till the end of the round. Then during every coming action stage he continues to try to roll a hit or critical hit till the card is flipped over or he is destroyed.
Umm, yes. That's how it would work, assuming that each round you chose to spend your action on trying to fix the guns. Of course, you may decide there's something you need to do more urgently - I had a ship fly around with this on it for four or five turns because I deemed Evade to be a more vital Action to take.
Or are you suggesting that you would get to roll for flipping the card AND take a normal action? In that case, no, you have to chose. It doesn't say Action Phase, it says Action in exactly the same way every other card that gives you a new option for your one Action a turn does.