My car would die right after starting. I do not know why.
I don’t kmow either . If you are not going to fix this yourself then find a shop you trust.
I am not a mechanic but I used to own a 1977 Corolla. I would suspect a fuel problem, maybe a vacuum leak. I don’t know whether the 1981 had a carburetor or a throttle body but either could be dirty. A good mechanic can fix it for you.
Chances are an 81 is carbureted. Many possible causes.
Plus an 81 still needs regular spark plug changes.
You need to take it to an experienced, independent shop.
Get an estimate to repair, that may cost you $100-150, the shop may waive the diagnostics fee if they do the repair.
But I would advise the estimate first, then decide if you want to keep this car.
We do not know the condition of the rest of the car, the chassis might be rusted to the point it might be unsafe. The suspension may be worn out. The catalytic converter might be clogged.
Only a hands-on inspection and diagnosis can tell.