The buses to new Sukhothai leave from the bus station about every hour during daylight. The bus station is across town from the rail station if that's the way you're arriving, you just take a tuk tuk or songthaew to get between the two. A trishaw out to old Sukhothai is about 150B from the bus station.
As far as I know, the only buses that pass Si Satchanalai historical park are the Sukhothai to Chiang Rai buses, they drop you on the road about 2 k from the main part of the park.
Thanks so much!