When you leave the arrival hall in front of the entrance there is a bus stop. You can find a ticket machine there. Go to bus nr 175 (heading towards Pl. Piłsudskiego). You will go all way to the last stop Pl. Pilsudskiego (duration approx. 38-45min). Then 5 min walk to the Old Town.
After getting off from the train go upstairs to the main hall. On the left hand side you will find a bus terminal. There get the 160 bus and get off on Stare Miasto Bus Stop (approx. 10-15min). Go up the stairs towards the Royal Castle and you are almost there.
After leaving the arrival hall get on the Modlin Bus. It will get you to the center of Warsaw in 58min. It will leave you in front of the Palace of Culture. Just next to it there is Marszałkowska street. Reach it to catch tram no 4 (direction Żerań Wschodni). Get off on Stare Miasto Tram Stop (approx. 10min). Go up the stairs towards the Royal Castle and you are almost there.