One of the FAQ-est of the FAQs. This is probably the most common problem
with any FI Honda.

See here:
http://www.tegger.com/hondafaq/startproblems.html#mainrelay

It is most likely your main fuel relay. I've got an '87 Integra and my
car started doing the same thing about a year ago and only stopped
when I fixed the main relay. Honda/Acura used some bad solder on the
relay so over the heating and cooling cycles the solder cracks and it
can lose connectivity when it gets hotter causing your fuel pump not
to go on. It is fairly easy to fix it if you've soldered things onto
circuit boards before. You basically have to remove the old solder and
reapply some new solder. If you don't know how then a local electronic
repair shop should be able to do it in 10 minutes for fairly cheap. Or
you could just buy a new one. The hardest part about repairing it is
removing it, it is up inside the dash above the fuse box and against
the firewall. It's a PITA to remove if you don't know how.
Here's a few good links:
http://techauto.tripod.com/mainrelay.htm
http://www.marklamond.co.uk/tech-honda/pgm-fi/main-relay/main-relay.htm
http://www.tegger.com/hondafaq/mainrelay.html

Also, the forums on www.g1teg.org are a great place to meet people
with the same car that may be able to help you if you're nearby.
There's also tons of resources on virtually anything about the first
gen Integra.