So the grammatical answer is that you repeat the verb in the negative.

• An rachaidh tú? - Will you go?
• rachaidh - Yes (just the future tense form of the verb go on its own)
• ní rachaidh - No (the future tense form of the verb with the negative particle before it)

You can give more verbose answers as well but that's the most terse way.

What they went with in the game is a bit more verbose to account for the fact that it was presumably not feasible for them to record a negative form of every verb that gets act of the player in the game and dynamically play them back at the appropriate time. It's the sentence "ní dhéanfaidh mé é sin" (I will not do that) said very quickly.