Is there public transportation between Cascais and Sintra?
Yes, I took the bus on that route in 2008 and rome2rio says it’s still running.
Yes, there are two buses that run between the two cities. The 417 and 403. The 403 goes by Cabo da Roca and is quite scenic part of the way. But, it takes significantly longer, going through tiny villages for the local people.
In high tourist season, both are pretty crowded. The website for the buses is Scotturb.
I took the 403 in May last year. If you want to enjoy the scenic beauty of Cabo da Roca, you need to get off and walk around - and then wait for the next bus. I didn't think the drive to Cabo da Roca was scenic, there were a few vistas from there to Cascais. The bus is a local, stopping often and takes a really long time. The coastal walk in Cascais to Boca do Inferno is beautiful doesn't take as long as the bus. Especially if there are two or more of you, take Uber in both directions, it's much less time and not expensive.
We took a taxi from Sintra train station to Cascais a few years ago, it was 18 Euros, about 20 minutes, if you're looking for a quicker way. We planned on taking the bus, but realized we were running out of time that day.