Broken carburetor air cleaner studs are common – and are easily repaired with a drill and an easy-out.
You’re going to need to use a power-tool, just do not put too much effort into it, or you can cause more troubles than you already have…
Finesse and plenty of penetrating oil like PB-Blaster goes a long way towards a successful repair. For safe measure, run a bottoming thread repair tap (not a standard tap) to guarantee good thread engagement – and prevent a pull-through. Here’s one I did on a Holley 4412-3 carburetor.