Seville is surprisingly close to the Algarve and the Portuguese and Spanish motorways are excellent, so it’s possible to get there in a couple of hours or so. The old centre isn’t that big, but there is a lot to see and you only have four and a half hours or so to do as much as possible. It’s well worth planning in advance and booking a small group guided tour of the Alcazar is highly recommended; you will see a lot more, and though you still have to pay the entrance fee, you can skip the long queues and save time.
I do recommend the day trip from the Algarve to get a taster of Seville, but be prepared for an enormous amount of time being wasted on the numerous hotel pick ups and drop offs. We had been on the coach for an hour just to get from our hotel to a petrol station in Albufeira! If the company rethought this aspect of the trip, it would be a much more enjoyable day out.
The description of the day trip is also highly misleading. This is not a guided tour of Seville, other than a quick ten minute walk into the centre and a brief orientation by the guide, padded out with a bit of a drive on the coach as you leave later in the day. You do not walk through the park or do the other things described on the website. Don’t expect these and you won’t be disappointed! You also need to be prepared for the commentary on the coach in several languages. This makes the journey time to Seville seem longer than it actually is!