Gambling in Fate
It's actually a thing with a history worth learning from! But before I start in on it, let me open with a warning: don't do The Poker Episode. You know the one. We're hunting down the ten secret keys to Dark Stobolous's planet-cracker all across the galaxy, this one crashed into the Chance Domes of Rota X, and in the opening shot it turns out that First Lieuteneral Twilliam has been sharping cards all over the breakrooms but after the closing credits we will never mention it again?
Fate Core assumes that the things that define PCs will stay relevant over the course of a campaign. If you take characters made without the knowledge that this card game exists or that gambling is a thing and throw them into a scenario where they have to engage with it, odds are none of their stunts will be relevant and none of their aspects will apply, and all you'll have done is make the game more boring for a little while.
Instead I'll be assuming not that, where people were aware of the cultural significance of gambling in the setting and made characters to that effect and somebody may even have written "Free-Spirited Card Sharp" in the high concept box or something. What's that going to look like?
A Skill Or Not A Skill
Back before Fate Core was even a thing, Spirit of the Century came out and billed itself as a game of pulp adventure, and the pulps were lousy with glitzy casinos and backroom games and fortunes turning on a single card. So they tried to drop in a Gambling skill.
This is a setup with an entire extra slant on the skill pyramid and about three times the stunts you get in Fate Core, mind, and they still had trouble making Gambling really stand up on its own as a thing you'd spotlight on a character. A followup book, Strange Tales of the Century, can't muster even a single new Gambling stunt, with all its expansion content.
Probably "not a skill" then. Fate Core suggested converting characters who were trained in skills that just dropped out, like Gambling or Leadership, to use aspects and write stunts that reflected the ways you'd use the remaining core skills to lead people or make bets. Contacts to find a likely gambler, Deceive to bluff, Empathy to read people, Notice to look out for tells, Lore to remember obscure rules, Provoke to force mistakes, Burglary to sleight some hands. (They provided examples in this wiki.)
You can use the idea of "aspects granting permission" here - a Free-Spirited Card Sharp is going to be able to use the existing core skills to gamble, casually, in a way other people can't. Or if you're in the Chance Domes of Rota X, where Everyone Bets Everything, anyone can gamble.
If you think you can make it work as a skill, more power to you; this advice is probably going to apply either way, but I wanted to get that out there.
The Casual Gambler
Spirit of the Century had this to say about setting up scenes to use its Gambling skill:
In practice, Gambling requires striking a balance between cool scenes and boring play. The moment that matters in a gambling scene is the last one, when everything is on the line and the last card gets turned over. [...]
Gambling scenes should usually be picked up in medias res. Give a quick rundown of who's at the table, making sure to include their body language, before picking up with the gambler character and the fall of the dice.
What's gambling for, though? In the casual case, getting stuff, same as Resources, but with a different spread of costs and benefits - there are things people will and won't gamble, and instead of temporarily being short on cash as a cost to failure, you can find yourself with a shady debt to a shady character.
You can use this to introduce various aspects of the game, the concept of side bets, et cetera, before whatever big gambling scenes you have planned. Even fairly high-stakes gambles for big pricey things can be resolved in a single roll - the drama's going to come in as Fate Points trickle into play and bonk off the various character and scene aspects in order to actually put up the necessary numbers.
The Big Gambling Scene
When you want to make gambling into the spotlight of the session, rather than a single dramatic story beat, the first thing you need to ask yourself is "what's the point?" I mean, maybe just "win the game" is the point? Planet-cracker key and all? But other things can be the point, too. Maybe the game is a distraction for a casino heist, and staying in for as long as possible is the goal, but winning would be pretty sweet too. Maybe the game is the point, but you can't win with Dark Stobolous's Doom Generator broadcasting misfortune waves and you need to hold on as long as possible while your team takes it out.
The second thing you need to ask yourself is "what's everyone else doing"? Besides First Lieuteneral Twilliam, I mean, who is through to the finals. This is going to follow from the first question. I really don't think side bets for side bets' sake are going to cut it, at this point.
Gambling, The Challenge
If gambling is one part of a multistep plan - say, Twilliam's out there on the casino floor being fancy while everyone else tries to track down and gain entrance to the high rollers' suite to bet on him for the planet-cracker - then use the rules for resolving a challenge. Give everyone something to do and Twilliam's part will be one gambling-related roll, probably Deceive but any of those earlier mentioned stand-in skills will likely do. Just keep in mind the basic rules of challenges - repetition is highly discouraged.
Gambling, The Contest
You can intersperse rounds of a contest with cutaways to anybody else doing anything else, so if you want to know if Twilliam actually wins anything while keeping up the pace elsewhere, you might consider that. Of the gambling-related skills mentioned earlier, Deceive, Provoke, and Empathy will likely make the relevant roll in the contest, and Notice (vs. Deceive), Burglary (vs. Notice), and Lore (vs. an escalating obstacle as relevant loopholes get more obscure) can be used to jockey for advantage.
Gambling, The Conflict
Or if you want things to work like a full-on contest of wills, you can just run a social conflict, straight-up, but expand the scopes of allowable attacks and defenses to reflect the fact that you can also stress out the opposition by being good at gambling.
You might want to run it in a wider scope that encompasses multiple successive hands of cards or even entire games in a single roll on Twilliam's part, giving people the opportunity to support them with Create an Advantage and such in between the big moments on the casino floor.