I can see why you are having problems.
Have just ran up a gazeteer entry for the route from Faro Airport and neither Google nor Viamichelin.com have updated their maps yet to take account of new roads.
The road onto the A22 motorway from Airport is reasonably straightforward.
1. Exit the Airport Car Park, follow the curve round to the right keeping in the right hand lane. Pass service station and roundabour on your left.
2. Proceed +- 3km down the N125-10 motor road till you come to a sign/slip road of to your right marked Portimão/Loulé/Lisboa.
3. Follow ramp round and carefully merge onto the IC4/N125. Continue along this dual carriageway for about 7km. After passing MSCAR showroom on your right (Just after traffic lights), get into the LEFT HAND Lane. Stay in that lane what ever happens.
After about a further 1500 metres, this lane will merge straight onto a stetch of motorway that after 3kms or so you will see the Algarve Footbal Stadium on your right.
Continue about another 2kms and you will see to your right, just after passing under a motorway bridge an off-ramp marked A22 Albufeira/Portimão/Lisboa. Take this ramp and merge onto the A22 and keep going till you get to the off-ramp marked "No. 1 - Vila do Bispo/Lagos"
From here there is a 8km motor road which will bring you to a roundabout on the N125. Turn left this will take you into the outskirts of Lagos.
From here follow your instructions to Porto de Mos.
