RegionJet is sort of the low cost carrier of trains. I havent heard anything bad about the service but I am on a wait and see mood with them. I have heard a few good things about customer service wiht MAV so I tend to be loyal.
RegionJet goes to Budapest-Kelenföld which is in Buda. Budapest-Kelenföld has a canopy over a few platforms I believe, but most of it is open to the sky (watch the weather).
From Budapest-Kelenföld to Pest you will either need a taxi for about 8.000 ft or you can ride the M4 which serves the station, then connect to the M3 to reach the center of Pest; or you can ride the 49 Tram or there is a bus that goes to Deak Ferenc ter in Pest (pretty sure). If it were me, I would take the 49 tram as it goes through some very interesting neighborhoods and great architecture along the way. I dont think Budapest-Kelenföld has a BKK office, but there are vending machines (must be ?) where you can buy metro tickets or you can use the BudapestGo App.
If not using RegionJet and you are departing Vienna use OBB. If departing Budapest use MAV. MAV will start and end at Keleti Pályaudvarin the heart of Pest .... well, okay, a tad off center of the heart but its a beautiful station and is well connnected with a short ride to any accommodation in Pest. BKK has an office down the front stairs at Keleti.