Connections between Portugal and Spain aren't good, especially when you're not involving one of the big cities.
Does your trip end in Alicante or elsewhere in Spain, or are you looping back to Portugal? If you pick up a car in Portugal and drop it off in Spain, you're likely to be hit with a really major surcharge that will make you forget that 5-1/2 hour drive.
I think TAP Air Portugal flies between Lisbon and Alicante, so that would be one possibility. What some other people have done, I believe, is get themselves over the border--probably to Seville--by bus and then move on from there by train. Of course, it's likely that none of them were heading to Alicante.
Rome2Rio lays out a bunch of options, none of them particularly attractive. Don't trust the travel times, frequencies or fares you see there. They are often crazy wrong. Just use the information as a guide to what sort of transportation might be available and where you might transfer. Keep clicking through the website and you'll eventually find the name of the company providing the suggested transportation. Go to its website for accurate schedule information.
Do you have a particular reason for choosing Faro and Alicante? They are not near the top of most visitors' lists.