Acura integra 1990 turns over fine but doesnt start

Am I right i thinking that you turn the key, the engine starts, but then when you turn the key back the engine turns off again and doesn't stay going?

If this is the case then it sounds like you have a worn out ignition switch. Not a particularly difficult or expensive fix. $45 will get you the switch and if you're savvy you can replace it yourself. A mechanic would take about an hour.

For you to check if it is the fuel pump, see if there is fuel running out of the hose if you remove the fuel filter. Another is to directly connect the pump to the battery. If the pump works that way, then it could be a efi relay issue which is found underneath the dash or a wiring issue.