Hi there,
I think the general rule here is to begin by fridging the Axolotl. Put him a container (ice cream tub) with fresh, dechlorinated, water, a secure lid and place him in a fridge.
This will help destress the Axie and will help relax his body.
Every day change the entire water, all 100% and keep doing this for a few days.
Don't worry about feeding etc for now, the fridging will slow the metabolic rate so that food is not required. You want the stone to be regurgitated and not pooped. Pooping could lead to full prolapse, not pretty and could be pretty fatal.
It's important you do this sooner than later, as the rock will gradually work through his system, doing it now may allow him to regurgitate the stone.
If is doesn't come out of his mouth, it may require a trip to the vets, as a stone that large could cause problems if he tries to pass it, their bodies are not a straight through system.
Please act quickly and let us know if you need any more help. Acting quickly, like you have, and doing all you can to reverse what has happened will most probably save you Axie. Keep us updated.
(on a side note ensure that once it's passed / removed you make the tank safe. NO stones or gravel of any kind. You'd be surprised how large a stone they can eat. If it's smaller than their head, it's edible.)