1991 Acura Integra RS - Starting Problem

I have a 1991 Acura Integra RS model with a 1.8Liter engine. I was having an intermitten starting problem with my Acura for about a year now and a couple of weeks ago it got to the point where the car not only wouldn't start every morning the first time I cranked the engine,after it ran for about 5 to 10 minutes the engine would shut itself down while I'm driving. I took it to a mechanic a friend of mine recommended and he found the problem(I was told by many part suppliers the problem was the fuel pump but the mechanic told me that Honda makes great fuel pumps and it is very rare to replace one). He discovered that when he got the car running and tapped the ignition key the car would shut right down. He showed me that when he kept pressure on the ignition key everything was fine but when he released the key the engine would shut down. The wierd part is that half the electric items would work and the other half were off. The radio would be on and you could crank the engine but the turn signals didn't work and he figured the electric fuel pump wasn't running either. I've been driving it for a week and no problems yet. My mechanic told me there is a relay that controls the electric fuel pump that he has seen go bad on Honda cars in case the ignition switch isn't a problem with your Acura.