Personally, I'd never do that particular Camino on a bike - it was crazy enough not getting hit by cars while walking with an orange kerchief held above our heads on a stick.
But if you insist, my advice would be to get a TALL VERY BRIGHT LOUD flag of some sort so people can see you when they come WHIZZING around those tight, very narrow curves where there is no place to walk, much less ride a bike.
I have walked a couple of the stages from Lisbon, I've made it as far as Azambuja. The Caminho through Lisbon itself goes on some, but not too many busy roads, and once you get to the Tejo river you have a few km along the river on a path. The off-road kms that I've experienced so far (from Sacavem to Alpiarte for instance, or the nice river path from Alverca to Vila Franca ) would be fine for cyclists. As would all the endless kms through industrial parks.
I am hoping that from Azambuja onwards I'll be in more picturesque territory, but I fear there are many kms along the national highways. I think that the problem here may be that the Camino in Portugal didn't get "resuscitated" till recently and there may not be too many off road paths in a lot of places.
BTW, I have found only one person in the towns in the 70-odd kms I've walked who has heard of the Caminho Portugues to Santiago, and she was the one in charge of Azambuja's Volunteer Firefighters' pilgrim accommodations (she told me, though, that the overwhelming majority of people she sees are walking to Fatima, not Santiago -- the routes are together from Lisbon to Santarem). Nor did the shopkeepers I asked on the path through Lisbon (whose stores were inches from yellow arrows) have any clue. I think that north of Porto there's more awareness (and even an albergue or two), but south you will find tons of yellow arrows but no one who knows what they are for.
I'm not trying to discourage you from taking this trip -- after all, I keep going back to walk a stage every chance I get! (hope to walk Azambuja to Santarem tomorrow in fact and I'll let you know). Some of the towns, like Alhandra and Alverca and Vila Franca da Xira are very pleasant, nice architecture, good rest stops. There are pretty stages along the river, too, from Estacao do Oriente (where it sounds like you're planning to start) to Sacavem, and then from Alhandra to Vila Franca da Xira. If you haven't seen my pictures from walking the first couple of days, here's a link http://picasaweb.google.com/laurie.reyn ... Dxipep3Vo#
Some more random comments:
-- If you look at the Gallego Amigos guide that's in pdf form on their website, you'll see lots of pictures of earthen trails, so I assume I'm going to find them at some point.
-- I can't imagine riding a bike from the Lisbon airport to anyplace. Do you have details on how you are going to do that?
-- I've walked the days from the Spain-Portugal border, and that stretch is beautiful, great accommodations, etc.
-- I think that south of Porto you are likely to be pretty much alone, and you're going to have to rely on private accommodations unless you take the Volunteer Firefighters' option. But the yellow arrows are everywhere!