BTDT, many times.

Run 12v and a good ground to the fuel pump so that it runs all the  time.
Start the car.  If it runs, find out why the fuel pump does not  stay on without
the jumpers.  Places to look: the pump fuse on the  fuse panel, the
electrical connections in the vicinity of the pump, the fuel  pump relay on the fuse
panel, the connections on the back of the fuse  panel.

Some relays have the fuse on top of the relay.  Check the fuse and  make sure
the connections to the fuse are tight.

The fuel pump electrical connections are covered with rubber but if the
rubber is cracked, moisture can get in and cause corrosion, high resistance, and
death to the connection.

The cover can be popped off the fuel pump relay and you can actuate it  by
making the relay connection manually.  If the car runs, the relay may be  bad
but you need to make sure the relay is getting the right voltage  signals.  A
volt meter will help.  You can also jumper around the  action of the relay by
pulling the relay and connecting the right socket  pins.  Don't have my book
handy now to tell which pins but you can figure  it out by looking closely at the
inside of the relay.  These are available  in junk yards and from list
vendors.

Most old Rabbits have had water leak onto the top of the fuse panel over  the
years.  This causes corrosion, etc., to the plugs on the back of the  panel.
Pull the panel down if all the easy stuff noted above doesn't  work and pop
one of the plugs off  the back and see what it looks like  inside.  If there is
rust and corrosion, good luck.  I got a good  used fuse panel from out west
off e-*ay for $15.  Solved all my panel  problems.  Most of the water comes in
through the antenna wire or through  the windshield seal, so fix that at the
same time.  Problems usually occur  when it is wet and cold and miserable out.

If you need to, you can keep the jumpers on the fuel pump and have it run
all the time (remove them when you stop the engine).  You can drive the car
this way for a while if you have to.  It is slightly more dangerous because  the
fuel is always under pressure even if the engine stops (as in a collision)
but not significantly more dangerous than driving a Rabbit in the first  place.

Let me know if you need more help.

Rabbits are good beaters.  They last a long time, are easy to fix  (after you
learn a few things), and the parts are cheap.  I got my Rabbit  Bentley off
e-*ay for <$20.

Tom

Rust belt Illinois
'80 Rabbit pickup, and Vanagons (RVC)