Everybody on Shadow Moses, except for the ones Snake killed, Ocelot, and Maryl were targeted by the FoxDie Snake was carrying to die... including Liquid Snake.
The only people FOXDIE was designed to kill were Solid Snake, Kenneth Baker, Decoy Octopus, Psycho Mantis, Vulcan Raven, Sniper Wolf, and Liquid Snake. (Of course, Snake killed Mantis, Raven, and Wolf before FOXDIE had a chance to.)
Liquid died from FOXDie so that must mean that at some point Snake will keel over from it to.
Naomi specifically states at the end of MGS1 that, unlike the other FOXDIE targets, Snake is set to die from FOXDIE at a random time. It could take effect in ten seconds, ten years, or he could live a full life with it never happening.
Snake is aging rapidly.
Technically, Snake is experiencing "accelerated aging."
When the project to make clones of Big Boss were starting Big Boss was in his late fifties
According to what Liquid said in MGS2, yes, but Kojima changed that aspect of the plot in MGS3. Officially, Big Boss was cloned during his forties, and Liquid simply had the wrong information. Not that it really changes anything.
Snake probably ages twice as fast as a normal person
It's unknown. The MGS2 script mentions that Snake had only begun to experience accelerated aging prior to the tanker incident. It could be that the accelerated aging occurs in bursts, or it could stop completely after a certain point. The "rules" for Snake's accelerated aging have never been clearly defined.