Bad Bad Move! Just looked on Google maps and the Hotel is right slap bang in the centre and the pedestrian zone. Not only will you have one hell of a time driving to the hotel through the very narrow streets, parking is likely to be like hens teeth and v. expensive. Also, on street parking with a Portuguese registered hire car is like hanging out a flag and saying "Come and get me!!!"
If I were you, I would park in the paid parking at Estacion Plaza das Armas or on Colon by the river and walk. Bear in mind that by mid-June midday temps may be approaching 35º to 40ºc. Distance about 500m according to Google maps.
I drive over to Seville regularly from Lagos and always do the above if I have to go into the centre.
