Transfers from Lisbon airport to Cascais
There used to be a bus operated by Scotturb but this was suspended in 2007 owing to poor traffic loads.
There are three ways to get from Lisbon Airport to Cascais.
1) Bus/Rail
There a bus service operated by Carris called Aerobus.
Information on the service can be found here:
http://www.carris.pt/en/index.php?area=servicos&subarea=servicos_tur_aerobus
This will take you from Lisbon Airport to Cais de Sodre Station where you can get regular suburban trains (every 15 to 20 minutes) to Cascais. Trip takes about 40 minutes.
2) Taxi - Lisbon - Airport to Cascais
To arrange a taxi, check with the Tourism Desk in the Arrivals Hall at the airport when you arrive. I would think you would be looking at +-€ 70, although prices rise at night and week-ends and baggage is extra.
2) Private Transfer
Low Cost Transfers are currently quoting € 36 for a transfer Lisbon Airport to Cascais.
See:
http://www.low-cost-transfers.com/lisbon-airport/
3) Hire Car
Some rental operators in Lisbon have daily car hire offers. Unfortunately, I can remember who, as it a year or so I saw the adverts when up in Lisbon. I don't normally recommend hiring cars if it is one's first visit to Lisbon/Cascais but it may be an option for you.