Does bullseye deal an additional 1 penetrating damage after the initial 2when throwing a light object, or does he only deal the one penetrating?

2, then 1.

If he uses it in a close combat attack, it's his base Damage of 3+1 for the object = 4, and then 1 Penetrating after that.

it goes damage from the object then the pentrating damage

ANY OBJECT A WEAPON: Bullseye can pick up and use light objects as if he had Super Strength. If he hits with an attack using an object, after the attack is resolved; deal 1 penetrating damage to his target.

it confused me a bit too, mainly because i didn't know if the meant the attack action itself of the attack roll

It means the attack (before the action resolves, but after the initial attack roll).

Between the resolution of the attack and the resolution of the action, there is this little window of opporunity for effects like this to occur. Most of the time, the resolution of the attack and the resolution of the action are indistiguishable from each other (so they seem to be the same). But in special cases like this, the difference between the two is important.