1986 Toyota Corolla Car only runs for 15 seconds!

My car suddenly wouldn't start. After much troubleshooting I determined that it is not the timing, fuel pump, etc. The car turns over just fine.

If the car sits for a few days, it will run (fire in other words) for about fifteen seconds. Then stall.

You have to let it sit for a long time before it will run again, but just for a few seconds.

My neighbor down the street said it seems to be a vacuum leak. While I was turning the key, he put his finger over some hoses + the car ran - I don't know exactly what he did. There are a lot of hoses under the hood - which exact hoses are the critical ones that I need to replace? I have a Haynes Repair Manual
that does not explain the whole theory very well.

Please explain what I need to do.