Last-minute trip to a friend's wedding in Portugal, so most of our weekend is booked but I've never been to Portugal (or actually Europe at all) so I'd love to see a little of it if I have time! We arrive from the US at 8 AM the day before the wedding and are planning to rent a car at the airport to drive to the wedding destination. Theoretically we don't need to actually leave Lisbon until around 2 PM to get there for dinner, what would do? I'm assuming we'll be jet-lagged and underslept and feeling like a walk and some sun would be in order, so I'm considering catching the Metro into the city somewhere and just walking and sightseeing and getting some breakfast/brunch before we head back to the airport to pick up the rental. Feasible? Or do you have any better suggestions?

We'll similarly have most of the day after the wedding in Lisbon - getting back to Lisbon around noon and our flight leaves very early the next morning, so all of the afternoon/evening for sightseeing. Was thinking of driving or training up to Sintra to see the castles and the park but not sure if we'd have time. Any input very appreciated!

You know yourself and you don't specify how long the drive to the wedding destination is, but driving after a long night flight is not wise. Yes, you could go into the city for a few hours, have breakfast and walk around, but by 2 pm you'll probably be exhausted. Again, you know yourself and how you behave when tired, you know how long the drive will be...just don't ignore these facts.
Sintra is not meant for cars, so if you decide to visit the last day get tickets in advance to skip the long lines and do it without a car. Even if you don't go to Sintra, you don't need a car in Lisbon.
For only a few hours, I think Lisbon would be a nice city to explore: Carmo convent, Santa Just Lift, Sao Vicente de Fora, a tram trip, plenty of things to see and do.

What xyz99 said

Thank you for the advice, this is all new to me! It's supposed to be about a 1.5 hour drive, the wedding is in Estremoz, where we will need a car for various planned activities from what I'm told. I'm hoping to sleep enough on the flight to be okay to drive but if I'm not I did already see there is a 12 or 2 PM bus we could take and I'll definitely switch to that option if it does not feel safe.

Carmo Convent and Santa Justa Lift were on my list from my searches, was thinking of doing Metro to the Baixa station where it looks like there are some luggage lockers and walking around that area. You're definitely helping me veto my friend's plan of driving the rental into Lisbon proper, it sounds like using public transit as much as possible is the way to go and no one has any specific sights they need to see (but we do need as many egg custard tarts as possible!).

The trouble is you may feel you are safe to drive but still aren't. Microsleeps can be fatal. New country, strange (manual gearbox) car, different rules of the road, navigating, all a lot for a tired brain to take in.
Aren't there others with cars at the wedding destination who can give you a lift if you get a bus there?
It will also make a tiny dent in your massive carbon footprint for the trip.

I agree with hetismij2 about the danger of driving after a flight through many time zones and the micro sleeps.

You can check all the buses from Lisbon Sete Rios to Estremoz for your date at https://rede-expressos.pt/en/timetables

There's one direct bus (w/o transfer) at 12:00 and another at 3:00. They both depart from Lisboa Sete Rios.

Another vote to forego the dash into town before heading out. Take the bus, reserve a car closer to the venue. Better safe than sorry.