How to get to Žeimiai: first of all, you need to get to Lithuania. Preferebaly to Kaunas but Vilnius is also not far away and there are more connections. For those arriving arriving on thursday and leaving sunday, there will be a shuttle bus from Kaunas to the rural mansion.
Cheap airlines fly to Lithuania. Mainly Ryanair and Wizzair. AirBaltic and Air Lithuania, LOT, and other airlnes are worth checking as well.
Kaunas airport is located 12km from the city center. The bus #29 takes about an hour to get to the city center (it makes all regular stops), costs €0.70.
Since 2013, Warsaw – Vilnius train has been cancelled due to the construction of RailBaltica project. The only option remaining, is to take a train from Warsaw to Bialystok, spend a night there, and then take a bus to Kaunas, which takes approx. 4.40 hours.
Other options by rail, if you come from this side, is Moscow, St. Petersburg or Minsk.
Locally, if you arrive in Vilnius, the trains are pretty fast and comfortable – it takes 1 to 1.5 hours from Vilnius to Kaunas by train (price range €4.63 – €6.37). Timetable and prices are here: http://www.litrail.lt/en
You may also take a train from Vilnius airport to Vilnius train station (costs €0.72)
Another option is public bus from the Vilnius airport to bus/train station (costs €1 from the bus driver).
There are many connections to get to Lithuania by bus. Some bus companies to check out:
Simpleexpress.eu, luxexpress.eu, ecolines.net, eurolines.com
Most of them will make a stop in Kaunas.
The bus will be provided to go from Kaunas to Žeimiai on Thursday night. If for some reason you willl need to arrive to Žeimiai at any other time, there are buses going from Kaunas to Jonava every 15-30min. (costs €2.19). From Jonava to Žeimiai the mini-bus goes few times a day (at 8.10, 10.00, 11.40, 16.00, 17.30). It costs about €1.