You can check the bus timetables for Helsinki-Vantaa Airport - Kivijärvi at http://tinyurl.com/3ekgq7k .

Timetable Jyväskylä - Kivijärvi is at http://tinyurl.com/3wwevo2

Combinations of trains and buses can be checked at http://tinyurl.com/3juasjt .

But as TampereMan suggests You cannot make the whole journey by public transportation on the day of arrival as the last connection all the way to Kivijärvi leaves around 3 pm.

By train you can get to Jyväskylä. A Pendolino train leaves from Tikkurila (a 25 min bus journey from the airport by bus #61) at 18:46, involves a change of train in Tampere around 19:56 - 20:05 and arriving in Jyväskylä at 21:36. From there you can continue only by a rented car or you can wait until the next day (which you could of course do in Helsinki, too).

We drove by car from Helsinki to Kivijärvi a couple of summers ago. If I remember right, it took us over 5 hours + a few stops for coffee & lunch. I'd take a train to Jyväskylä in the evening and stay the night in Jyväskylä. You can rent a car there too, if you like. Saves you the tedious drive Helsinki -Jyväskylä.

Kivijärvi has (virtually) no local transport btw.