There are wifi hotspots in the Algarve but not in Carvoeiro.
Best is Mc Donalds in Portimao. FREE.
If you are not living in the Algarve for more than 6 months of the year it is probably cheaest to use the Internet Cafe's.
If you want your own setup the cost is around 20 eours per month but it is a minimum 18month contract with TNM or Vodaphone. Ask for the broadband system at Algarve shopping or Forum Algarve ETC.
The pool at Carvoeiro Sol is sort of heated but it seemed dirty to me.
Most kind regards,