To go to Rundale you first take a bus to Bauska, they run every 15 minutes,(1 hour) Be sure that the latest you take is 9.40, try to be earlier. The fare is 2,15 Lat and you buy a ticket inside the busstation
In Bauska there is a bus to Rundalepils (castle of Rundale) at 11.00 o'clock, the next one is at 15.00. If you miss the first one you can take a taxi, they are waiting on the other side of the bus station building or like we did go hitchhiking. We walked back to the turnabout where I saw the sign Rundalepils, from there it is 11 km.
On my photo you can see the timetable for the bus back from the castle to Bauska.
To get to Rundale Palace from Riga by public transport, you need to take a bus from Riga's central station to Bauska and then connect to a bus to Pilsrundale (not Rundale which is past the Palace).
There are dozens of buses going from Riga to Bauska, check the schedule on the attached website or look at the schedule at the bus station to see which bus is quickest, the times range from about 1 hour to 1:40 depending on the route.
There are only a few per day traveling from Bauska to Pilsrundale so check the schedule before you go (I think Bauska-Eleja-Jelgava is the route we took). Alternatively you can take a taxi from the station to the palace and then take a bus back.
The fare was 1.45 lt each way (1.2 from Riga to Bauska, .25 from Bauska to Pilsrundale). You can buy on the bus or at the ticket window in the station. Try and have small or exact change.
Our timing was perfect, the bus ride from Riga to Bauska took 1:10 (caught the 9:40, arrived at 10:50), the bus from Bauska to Pilsrundale took a little over 15 minutes (caught the 11:00). Look for the sign in the picture, after you get off the bus, head across the street in the same direction the bus was going, the entrance is not far from the bus stop.
Check the bus schedule on the sign across the street for return times, the soonest one we could catch was 2:30.
Take the bus from the Riga’s central bus station to Bauska. In Bauska you have to take bus to Pilsrundale. Buses to Pilsrundale from Bauska are every 2 hours.