If at the beginning of my turn my icons joker (without Armor piercing) is near to a figure with invulnerability can I outwit invulnerability to deal 1 damage with poison?

Yes, you can. It was ruled that Outwit can be used before Poison's effects come into play.

K-ness is correct.
To go into more detail, Outwit can be used at any time during your turn, including at the beginning. When two or more effects happen at the same time, you get to choose what effect happens first, so any effects that can happen at the beginning of the turn can be resolved in any order - for instance, it is fully legal to Outwit a damage reducer, deal Poison damage and then roll for Leadership.