The buses from Liverpool Street were indeed the main way of getting to and from the airport (they later ran from Canning Town) and were entirely free. They really wanted you to get out there in those days...
Not only with Crossrail / the Brenda Line / whatever not call at LCY, none of the nearby stations will be within a direct rail link of the airport, either. Both Canary Wharf and Custom House are on different branches of the DLR to LCY, and Woolwich is a different station to Woolwich Arsenal. Oh well...
I wasn't around in London in the 90s, but do miss the airport's early 00s days, when I would cycle up to the terminal (oh yes), wander up to the check-in desk with 10 minutes to spare, they'd have your BP pre-printed and know your name (good old KLMuk as it was at the time) and you'd be through security and on the plane in a couple of minutes. Hard to imagine that now.