Ours for the Taking reads: "Any Phase: Choose 1 location that would kneel to trigger its effect. Copy that ability (paying any additional costs),without kneeling the location."

I have Longship Foamdrinker (Limited Response: After you win a challenge,kneel Longship 'Foamdrinker'; to choose and kill a participating character. (Limit 1 Limited Response per round.)) in play.

Can I use Ours for the Taking to copy the "ability" of Longship Foamdrinker if I have already used a Limited Response this round? I'm confused as to what the "ability" of the card is...

Ultimately, you can because it is OftT's "Any Phase" effect that is resolving, not a Limited Response.

Ok, but since we copy the whole ability, what with "after you win" requirement?

That is correct. Because it is the Any Phase effect that is triggered and resolved, you will never be able to meet the play restrictions of a Response effect. You could still use the event to copy the location's ability, but that ability will "fizzle," so it isn't advisable.

So, actually the "Limited" part doesn't come into it. Copying any Response ability with Ours for the Taking isn't going to work out well.

Correct. Unlike something that tells you to copy the "effect" of a card, Ours For the Taking tells you to copy the entire ability. That means play restrictions, costs, etc., are all copied along with it. The "after you" play restriction on a Response will almost never be true when you play the event, so copying a Response ability will usually be an exercise in futility.