Would an atheist wanting to pray be irony?

No. That's a contradiction, but not irony. Irony is when there's a twist. If a religious tennis player prays for the win, but the atheist wins, that's irony. 

In Alanis Morissette's song "Ironic", the examples are coincidences, not ironies (which makes the song ironic). She says "Rain on your wedding day..." but that's just coincidence. It's not necessarily unexpected. If, however, you plan your outside wedding for Las Vegas because it never rains there, and then it rains, THAT's ironic, because you could have saved your money and had it near home.

No, it would only be a contradiction.

There has to be an ironic 'twist'.

Ironic: "Happening in the opposite way to what is expected, and typically causing wry amusement because of this." 

Ironic would be the atheist dies and goes to heaven, 

but his friend a devout christian goes to hell.

Nah - just ridiculous.

wanting - no. Praying to get something, save his life (etc) - yes. ["There are no atheists in foxholes."]

An atheist could want to do that just for mock a religious believers ¯\_(ツ)_/¯

I'm not an expert but I don't think it's irony, someone can crave a feeling of something they won't attain. I really wish I could have a low anxiety day, i know it can't happen but I still want it. I believe in God, but i think this is how and Athiest would think

yes it would because they do not believe in any higher power and therefore praying would be considered irony.