Sintra as a starting point?

I considered doing it this way myself for about the same reason when I flew into Lisbon in May. I wound up day tripping to Sintra after all, by train.

I did sort of regret not having a night in Sintra - it seems like a peaceful town or would be at night, anyway. But it was more of a "If I had an extra night, might have been nice to spend it in Sintra" not "wish I hadn't done this as a day trip!" I don't think it's a mistake to start overnight in Sintra, though.

One downside of this approach: you are locked into spending time in Sintra on those particular nights. If you day trip from Lisbon, you can do it any day you want; get a rainy day in Lisbon and you can spend it in museums and go to Sintra on a nice day. Also, I understand that Sintra is busiest on Monday (many tourist sites and museums in Lisbon closed) and on weekend (when locals head to Sintra), for whatever that's worth.

The train is really easy. You can take the subway from the airport to Oriente station and just take the train from there.

If you arrive in Lisbon at 7:00, allowing an hour to disembark, pick-up luggage get to the Metro; then an hour to get to your hotel or alternatively the train station (probably more like 1-1/2 hours), it's about 9:30. Assuming your timing is just right and you get the 10:01 train arriving Sintra 10:41; then allow a half-hour to drop your luggage at your hotel (pre-booked), it is then around 11:30. From what I've read and learned on the RS Forum, this is when the day-trip buses from Lisbon arrive, so Sintra gets busy. The research I've done indicates the palaces close between 17:30 and 18:00.

So, it will be 4:41 am Boston Time and you'll start your day visiting palaces and Castles. Admittedly, we're in our 60's but for we old-timers, and many other folks, this will be a tiresome visiting period. If you have lunch (Sintra time)/breakfast (Boston time), baefore or between visits, from what I've read you may achieve admission to 2 venues.

You noted you can't check-in to your hotel in Lisbon until mid afternoon. Do you already have a hotel booked in Lisbon? If so, can you change your plans (to stay in Sintra instead) without a financial penalty? If you haven't booked your hotel, check around and see if you can't find a place with early check-in. I have to believe in a city the size of Lisbon, you can find accommodation that you could check into by 11:00 at least. That way you'd only have an hour to "kill" having a bite to eat, before checking in and getting a little shut-eye. (remember it's still only 4:00 am Boston time, Then, 15:00 hr (or so) Lisbon time, get up and begin your visit of Lisbon, perhaps heading for Alfama and the Castelo de Sao Jordge, Museo de Artes Decorativas, perhaps attend a performance at the Museo do Fado, or walk down to the Se. Then have dinner (Lisbon time) and head back to your hotel. Next day, or the day after, head to Sintra (make your choice on weather conditions and not going on a weekend if possible).

If If you've not been to Lisbon previously, 2 nights isn't much time (assuming you spend 2nights in Sintra). Also, if you stay in Sintra and Lisbon you'll be spending time checking-out and checking-in 2 hotels, and dealing with luggage twice. Admittedly, my spin on your question is driven by the limitations of "logistics" and "jet lag", not by opportunity of remarkable venues there are to be seen in Sintra (even if some will be seen through, arguably, bleary eyes).

Have a great time in Liston and Sintra.

Last year, I did pretty much what you are considering, and it worked out wonderfully. I flew from California, connecting through Heathrow to Lisbon, then got the Metro to the train to Sintra, and was quickly in Sintra to spend 2 nights. I had an afternoon/evening, then a full day, then another morning in Sintra, before heading back to Lisbon. I could have easily enjoyed another day in Sintra. I stayed at a great B&B which was a few minutes walk from the train station, and then a nice 10-15 minute flat walk to the center of town, traditional and modern public art, and parks/gardens, along the way. In addition to an enjoyable town, there are 4 or 5 great sites, palaces or estates generally with expansive gardens, and ruins of a castle above the town, each of which easily merits a half day. Also some excellent restaurants and bakeries around town. I have also heard that Mondays are busy in Sintra, as so many museums and sites are closed in Lisbon on a Monday, so it is a popular day to day trip to Sintra. Of course both Lisbon and Porto/Douro Valley deserve visits of several days as well. Enjoy beautiful Portugal!

With only four nights in Lisbon, I would not spend two of those in Sintra! But, I’m not overly fond of Sintra (I realize I’m in the minority). Most any hotel will allow you to drop off your luggage if you aren’t able to check in, which eliminates that issue.
There is too much to do and see in Lisbon to only spend two nights there! And, spending a relaxing day after your flight just walking around Lisbon, having a nice lunch then a short nap, is much preferable to me than fighting the crowds in Sintra. And, remember, any time you change hotels, you lose a good portion of a morning or afternoon packing, trekking to the new one, checking in, unpacking, etc.
A taxi from the airport to Sintra would likely be in the 50€ range, as it costs me 35-40 to Cascais.