There are 4 options:
1. taxi, about NT1300, maybe more with holidays
2. private car, i can recommend driver/mercedes for NT1100 [pm me]
3. bus - NT$150, clean, safe and reliable and you do have time for this option. straight shot from CKS airport to Songshan airport, runs regularly [15-30 minute intervals]
4. shuttle to train, then taxi when you get to taipei - that would be last on list of choices.