Is there a possibility to go across the border with a rental car?
Yes, check here:-
To view the CROSS BORDER or FERRY terms and conditions. At STEP2 of our 3 step online booking process, alongside the car photo of your choice, there is a link entitled “IMPORTANT INFORMATION”. Clicking on this link will open an additional smaller window, with the terms and conditions appertaining to that particular car/supplier. Choose then the CAR tab and look under TRAVEL RESTRICTIONS.
Try a few different suppliers as each supplier/vehicle type may have a different cross border rule.
Are there any additional costs to Portugal?
Not normally for there.