After eating something that didn't sit well, cats are rarely unwell for longer than 24 hours. We would expect that the worst of the vomiting/nausea has now passed and he should improve quickly with time.
To avoid any more vomiting, it would be best to offer very bland and digestible food little and often for the next 2 days or so. Small meals of either boiled white chicken or boiled white fish and rice should be fed as are often not brought back up. Typical cat food may be too rich for an unsettled stomach,
If he is choosing to not voluntarily drink, a few tablespoons of water should be added to the meals so we know he is drinking something.
With sticky gums, the worry would be that he is a little dehydrated. We would thus be very keen to have him eat or drink something ASAP. Warming the food can help to release flavours and smells and encourage eating. Similarly, he may accept the food from your hand rather than a bowl. I would try to get something into him now before bed time.
Given his age, if the vomiting has not stopped and he has not picked up by tomorrow (or deteriorates in any way), a vet check would be advisable. This would mainly be to ensure he is not dehydrated, but also to assess if there is anything else going on. They will do an all over check and will certainly have a good feel of his abdomen to ensure there is no discomfort or sign of an obstruction. They will check for other things such as a fever. If they have any concerns, they may decide to run some tests, such as some blood tests.
The vet would be able to give some anti-nausea and ant-acid medicines to settle his stomach so he can keep food down and start to recover. If it turns out that he is dehydrated, they may decide to give him some supplemental fluids.
Please let me know if you have any further questions.
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