I flew in and out of Faro two years ago and same with Malaga three years ago both times from Dublin and on Aer Lingus with large plastic bag (airline supplied) on outward journey and un-bagged or boxed on return.
Malaga: there's a Decathlon store about 15 minutes cycle away from the airport in the estate on the opposite side of the main road that runs past. You should be able to get a bike box there but advise contacting them ahead of departure. Good medium sized airport.
Faro: It's quite close to the sea and when I flew in I had arranged a Warmshower (
https://www.warmshowers.org/ )stay with a host on Praia de Faro which is only about 20 minutes cycle away over the bridge. I then the next morning went back over the bridge and turned left through a nature reserve and then inland after a couple of kms. through one of the most exclusive residential areas in Portugal built round a golf resort. I know there's a bike rental shop in Almancil nearby because I bought a spare tube there.

Praia de Faro

cycle path through nature reserve

House in Quinta do Lago golf resort.