Gaskets don't just fail spontaneously. Guessing it overheated so you'll have to solve that or any engine repair is wasted money. Coolant pump, thermostat, radiator, engine's coolant passages, hoses - there's a problem somewhere that you'll have to fix. Aluminum heads don't tolerate much overheat. It's probably warped and will need machining, although there's a limit to how much can be shaved off so you may be up for a new head.
Guessing he means one cylinder is losing compression by leaking through the failed gasket, not that there's a cylinder/ring problem to address.
Get a quote. Be clear that the quote is the quote, no room for "oh we had to do more than we thought so you owe us an extra $2000." Just be honest that you only have so much to spend and need to compare repair cost to replacement. So ask what it's going to cost for repairs and the likelihood of those repairs being needed. He'll give you something like "For X this many dollars. For Y, which will probably also need to be done, this many more dollars. For Z, which might need to be done, this much." Be clear that you don't have money for surprises. You have to know the worst case cost before proceeding. It's perfectly reasonable to ask for all possible costs so you know what you
might be spending. It isn't reasonable to expect him to know exactly what it will cost. He can't know, for example, if the head is warped without taking it off and examining it, only then being able to determine if it's true, needs machining, or needs replacing. But he can give you the costs associated with each possibility up front.
Compare it to the cost of an engine replacement or another car. If you haven't taken particularly good care of this one then this might not be your last mechanic's bill, so factor that in to your decision.
Imo vehicles should have a thermocouple fuel shut off. There's one on my air compressor yet none on my vehicles. Anyone whose car has overheated will probably agree. But I guess that would present an insurmountable 'safety issue' somehow.

Not like when an engine turns off anyway because it died, right?