The "S" light is the car throwing an ECU or TCU code. To check this,
there is a two prong connector behind the passenger side kick panel
(mine is blue). You must remove, or pull back the panel, find the
connector, and jumper the two pins with a paper clip, and then turn
the key to the power on position (do not start). You will get a series
of flashes from the "S" light. A long flash counts for 10, a short
flash counts for 1.  for example, one Long flash followed by seven
short flashes  is a code 17 commonly showing Vehicle Speed Sensor
which was my problem. Use google, or a Haynes or Chiltons manual to
match the code to the problem. Keep in mind it may be multiple
problems.

The power not coming on the way you are describing sounds a lot like
the main power relay, or so I have heard. I haven't had to change
mine, but have heard of several people describing the same thing. You
may want to Google "91 Accord Main Relay" and look over some of the
results to see if any of the symptoms people mention under starting
problems match up. From what I've read, failure of  the main relay is
fairly common in 90 - 93 Accords. You can Also get good information on
how to change it as well.

Most likely your problem is the main/fuel pump relay. Located on the
left side of the dash above the cruise control ecu. Usually the relay
will act up
when the vehicle is hot inside