I live in Lisbon, but I don't take taxis so I can't help you there. An uber from the airport to the city centre will cost you ~€12 in normal conditions. If you change your mind about the metro, it's a good system. To get to where you're going in the Baixa, you'd need to take the red line (direction São Sebastião) to Alameda and then switch to the green line (direction Cais do Sodré) and get off at Baixa/Chiado.
Boa sorte e bom caminho.
Kind of an off-topic question, jungleboy, but do you think the 12 euros you quoted is a standard Uber fare to the center? If so, it’s really not much more, if at all, than the normal taxi fare. I remember being in Lisbon a couple of years ago when thousands of taxi drivers blocked the streets in protest of Uber, but I never heard what resolution there was of that, if any.
I am a huge fan of the Lisbon metro, but the ride out to the airport is almost farcically circuitous, don’t you think? I personally would have extended the line from Campo Grande, but no one asked me, unfortunately.
P.s. I think the aerobus runs 24 hours, doesn’t it? That might be the better option than the metro for a cheap way in