1) I notice that both Budapest & Bucharest now have heavy rail links to their respective international airports. I have found times out quite easily for Budapest (via Ferihegi) but not so for Buchurest. Does anyone have a link for the schedule?
The information you want
is here, respectively
here for the PDF timetable. As you can see, the schedule is a bit hit-and-miss though, with several gaps between trains longer than an hour.
Additionally, they haven't actually built any additional infrastructure, but simply reused the existing Bucharest - Urziceni railway line, which means that the Airport station is in fact about 2 km away from the airport itself.
So if you don't want to walk, you have to use the connecting shuttle bus (which is included in your ticket), whose schedule is heavily padded, though I can't judge whether they are just being overcautious, or whether train delays and road traffic conditions are really that bad: 15 mins of connectional allowance, and 20 mins for the 2 km drive to the airport.
Alternatively, RATB operate two express bus lines from the airport to the
main station (780) and the
city centre (783). Service intervals are definitively better there, but the problem there is that they don't give you any indication on how long the journey is supposed to take.
If I remember correctly, it should take you approximatively 30 mins from the airport to the station (but don't quote me on that

), which would make it faster than the rail service, but day-time traffic in Bucharest can be quite bad, especially during peak hours, and it might well be that the bus gets stuck in traffic jams.