1990 golf sputtering and dying

golf hesitates at lights and then is fine when accelerating. it died the other day and we could not get it started again. we tried for an hour while waiting for the tow but it would not catch. Had battery power for the starter. When i started pushing it off the road it started up. WHY??? What should i look at under the hood?

start with basics. does the car get: air, gas or desiel, and electricity?