There is no direct easy route by public transport from Seville to Evora.
If you are flying into and departing from a Spanish Airport, there should be no problem taking a Spanish Hire car into Portugal. (Provided you inform them upon hiring the vehicle.)
The following guys have a programme where you can even pick up a car in Spain and drop it off at Faro Airport in the Algarve for a reasonable cost.
http://iberiacarhire.com/
As regards buses, your best option would probably be to get the daily DAMAS/EVA bus from Seville to Faro as follows:
Dept Seville: 07:30
Arr: Faro 10:10
then the Rede Expressos departure as follows:
Dept Faro: 11:15
Arr Evora: 15:45
