I think that 270 isn't bad for having a taxi at your disposal for 8 hours ... it means you & your sister won't be slowed down by your dad's understandable lack of stamina. But as you say, Dad likes planned stuff, being taken care of, not big on spontenaity. There are some excellent private tour guides, who charge around €200 for several hours, taking you to Acropolis, and whatever sites you want. However, it's mainly walking unless specified. It's hard to advise you when you don't share more about his preferences. Will he insist on going up to the top of Acropolis, it's a steep climb, or would he be OK with sitting in a cafe? Is he paying for all this? That also matters. Whoever pays the piper calls the tune! Some options:
(1) Full-day taxi tour at 270€, dad can get out of car or not, depending on his interest & stamina.
(2) Private LICENSED guide, with/without car, sample of services & costs, here's a highly recommended one
https://tourguides.viator.com/tour-guide-diana-stefanou-huttinga-1752.aspx
(3) Taxi from cruise dock to Herodion (40E). Check in, refresh, quiz dad on his interests. If he's OK sitting on balcony or roof terrace with a beverage, you & sis run off to scale Acropolis with R Steve's D-I-Y audio on your phones. Return & take Dad next door to Acropolis Museum for walk-thru, and nice lunch on outdoor terrace looking up at Parthenon. 2 of you or all 3 -- wander thru Ancient Agora, check out its small museum, photo ops. THEN be picked up at your hotel for a 3 or 4 hour taxi sightseeing drive around the other Athens highlights ... you can google "Athens 4-hour taxi tour" and get dozens of such, with reviews. Many have hourly rates like 40-45€. YOU will dictate what you want to see ... but that means you'll need to do a little homework ... browse thru a good website, print out an Athens Map -- here's a simplified one, http://www.planetware.com/tourist-attractions-/athens-gr-ath-ath.htm#GR107 - just showing tourist highlights.
(4) Same as #3, through the lunch. Then Pick up the Happy Train! http://www.athenshappytrain.com -- In athens I do NOT recommend the hop-on/hop off BUS, because it's limited to big streets & you end up sitting in traffic. But the "happy train" can go on small lanes, and no-car streets, and is enjoyable. and you can get off a couple of times & re-board the next one. This may be the "Goldilocks Version" for you!
CORINTH -- No. Unless you are well-versed & ancient history buff, hard to get the most out of this site... and it's a long trip out of town. Some religious people go there to stare at a stone that St. Paul stood on to preach ... but you can do that right in Athens, looking at the Areopagus (Mars Hill) right next to Acropolis, where Paul preached as well. https://en.wikipedia.org/wiki/Areopagus