I took a bus from Hanoi to Laos back in 2011.
The bus took around 23 hours. This includes maybe an hour at the border as most tourists needed to get Laos visa. The road taken was through very steep and dangerous looking mountains. The bus had sleeping berths in it rather than seating ones so that was a big plus. Annoying thing was that about an hour into the journey as mostly drunk passengers where beginning to drift of to sleep, the driver decided to play some telenovela with load kpop music over it. Loud shouts of "STFU" and local insults have not made him change his mind about music selection or the volume. Bus driver has likely been hitting ya ba balls as he seemed rather jittery. It was a night bus so not much sightseeing.
I would guess that the reverse bus from Laung-Pra Bank to Ha Noi exists as this guy had to drive back eventually. Possibly also goes further to either Vang-Vieng or Vientiane before returning. Whether this is the case now I do not know. Although both those countries have had major spike in internet users and services offered online over last few years, it is possible not to find anything about such bus unless locally present.
Laos has total of 4 km train track so, no trains except short stretch to Thailand.
Also would like to add that you should ALWAYS ask for ticket unless you want to pay twice.