The problem very likely is the notorious main relay, which
ultimately switches power on to the fuel pump, a problem for
countless circa 1990 Honda owners. It's located near the
driver's left knee, somewhat high in the dash. This relay
has electrical joints whose solder weakens with heat
cycling. Yours are the classic symptoms. Try some of the
hints at the following web site to confirm and fix (probably
for well under $100):

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